Inhibition of IMPDH from Bacillus anthracis

Inhibition of IMPDH from Bacillus anthracis: Mechanism revealed by pre-steady state kinetics Yang Weia,1 , Petr Kuzmiˇcb , Runhan Yua , Gyan Modia,2 , Lizbeth Hedstrom∗,a a Departments

of Biology and Chemistry, Brandeis University, Waltham, Massachusetts b BioKin Ltd., Watertown, Massachusetts

Abstract Inosine-5’-monophosphate dehydrogenase (IMPDH) catalyzes the conversion of inosine-5’-monophosphate (IMP) to xanthosine5’-monophosphate (XMP). The enzyme is an emerging target for antimicrobial therapy. The small molecule inhibitor A110 has been identified as potent and selective inhibitor of IMPDHs from a variety of pathogenic microorganisms. A recent X-ray crystallographic study reported that the inhibitor binds to the NAD+ cofactor site and forms a ternary complex with IMP. Here we report a pre-steady-state stopped-flow kinetic investigation of IMPDH from Bacillus anthracis designed to assess the kinetic significance of the crystallographic results. Stopped-flow kinetic experiments defined nine microscopic rate constants and two equilibrium constants that characterize both the catalytic cycle and details of the inhibition mechanism. In combination with steady-state initial rate studies, the results show that the inhibitor binds with high affinity (Kd ≈ 50 nM) predominantly to the covalent intermediate on the reaction pathway. Only a weak binding interaction (Kd ≈ 1 µM) is observed between the inhibitor and E·IMP. Thus the E·IMP·A110 ternary complex, observed by X-ray crystallography, is largely kinetically irrelevant. Key words: enzyme kinetics; inhibition; IMP dehydrogenase; B. anthracis; stopped-flow; steady state Abbreviations: CBS, cystathionine β-synthetase; CV, coefficient of variation (%); IMP, inosine 5’-monophosphate; IMPDH, IMP dehydrogenase; BaIMPDH, IMPDH from Bacillus anthracis; BaIMPDH∆L, BaIMPDH Glu92-Arg220 deletion mutant; CpIMPDH, IMPDH from Cryptosporidium parvum; XMP, xanthosine 5’-monophosphate;

1. Introduction Antibiotic resistance is a world-wide problem threatening the effective treatment of infections caused by pathogenic bacteria [1, 2]. New antibiotics and targets are urgently needed [3]. It is also important to develop new antibiotics against potential bioterrorism agents [4], such as Bacillus anthracis, the causative agent of anthrax. Inosine-5’-monophosphate dehydrogenase (IMPDH) has recently emerged as a promising antimicrobial drug target [5, 6]. IMPDH catalyzes the oxidation of inosine-5’-monophosphate (IMP) to xanthosine-5’-monophosphate (XMP) with simultaneous reduction of the cofactor NAD+ to NADH [7]. The conversion of IMP to XMP is the first and rate-limiting step in guanine nucleotide biosynthesis pathway. Inhibiting IMPDH causes an imbalance in the purine nucleotide pool that suppresses proliferation. The reaction involves the initial attack of the active site Cys on C2 of IMP coupled to NAD+ reduction, to form a covalent intermediate. NADH departs, a disordered flap folds ∗ Corresponding

author address: Sanford-Burnham Institute, La Jolla, California 2 Current address: Indian Institute of Technology, Banaras Hindu University, Varanasi 1 Current

Biochemistry

into the empty cofactor site and the covalent intermediate undergoes hydrolysis. Several inhibitors of mammalian IMPDHs, most notably mycophenolic acid, bind selectively to the covalent intermediate and prevent hydrolysis [8]. Selective inhibitors of IMPDH from Cryptosporidium parvum were identified by high-throughput screening followed by structural refinement [9–15]. These compounds effectively inhibit CpIMPDH activity in vitro, display significant antiparasitic activity against an engineered Toxoplasma gondii strain relying solely on CpIMPDH, and show a therapeutic effect in a mouse model mimicking acute human cryptosporidiosis [9–12, 14]. Some of these compounds are also effective against IMPDHs from bacteria such as Bacillus anthracis and Mycobacterium tuberculosis, suggesting their potential use as antibiotic agents [5, 6, 13]. The triazole compound A110, which inhibits BaIMPDH with IC50 = (43 ± 3) nM [15], displays in vitro antibacterial activity against B. anthracis and Staphylococcus aureus [13]. Related “A” series compounds are uncompetitive inhibitors with respect to IMP and noncompetitive (mixed) inhibitors with respect to NAD+ . These observations suggest that A110 binds after IMP, but do not reveal whether it binds to E·IMP or another downstream complex. Crystal structures of small molecule inhibitors complexes with BaIMPDH and Clostridium perfringes IMPDH have been solved, indicating that A110 binds to the cofactor site. However, these structures may not represent the highest affinity enzyme-inhibitor complexes [9, 15, 16]. In this report, we utilize stopped-flow rapid kinetics techniques, supported by confirmatory initial rate experiments, to investigate the detailed inhibition mechanism of the

Ms. No. bi-2016-00265u.R2 • Revision 2 • ACCEPTED FOR PUBLICATION • August 19, 2016

small-molecule inhibitor A110 against a subdomain-deleted form of the B. anthracis enzyme, BaIMPDH∆L [15]. The results show that A110 binds predominantly to the covalent intermediate E-XMP∗ . We discuss these kinetic results in terms of possible conformational and structural effects. We also report the values of microscopic rate constants that characterize the catalytic cycle of BaIMPDH∆L. The mechanistic insights should prove useful in the rational design of IMPDH targeted therapy.

replicated twice, as part of sessions 1 and 2. The experiments with added A110 were also replicated twice, as part of sessions 3 and 4. Thus the combined stopped-flow data set consists of 6,720,000 raw data points, resulting in 560,000 averaged (n = 12) time points, organized into 56 kinetic traces (eight replicated series each containing seven kinetic traces observed at [NAD+ ] = 0.25 – 8 mM). Each combined group of kinetic traces, obtained while simultaneously varying the concentrations of NAD+ , NADH, and A110, was subjected to global regression analysis [19]. The nonlinear least-squares regression model for individual kinetic traces is defined by Eqn (1), where A is the absorbance (in mOD, 10−3 absorbance units) at 340 nm at reaction time t > 0.004 sec; A0 is the adjustable baseline offset at t0 = 0.004 sec, essentially a property of the instrument; rQ = 6.22 mOD/µM is the molar response coefficient of NADH; [Q] is the concentration of NADH at time t; and [EPQ] is the concentration of the ternary complex E-P.Q. It was assumed that the UV/Vis extinction coefficient of NADH (6.22×103 mM−1 cm−1 , [20]) does not change upon binding to the enzyme and therefore the extinction coefficients of NADH and the enzyme complex are exactly identical.

2. Materials and Methods 2.1. Materials IMP disodium salt was purchased from MP Biomedicals. NAD+ free acid was purchased from Roche and NADH disodium salt was purchased from Acros. Compound A110 was synthesized as described [17]. DTT, TCEP and IPTG were from Gold Biotechnology. All other materials were from Fisher. 2.2. Experimental Methods 2.2.1. Protein expression and purification The construction of the plasmid expressing His6 tagged BaIMPDH∆L was described in [15]. The plasmid was transformed into E. coli BL21 ∆guaB competent cells [16]. Histagged protein was over-expressed at 18◦ C for 20 h, and purified with Ni-NTA Sepharose beads (GE) at 4◦ C in lysis buffer (50 mM phosphate buffer, pH 8.0, 500 mM KCl, 5 mM imidazole, 1 mM TCEP and 10% glycerol). After elution with 500 mM imidazole, pure protein fractions (> 95% as analyzed by SDS-PAGE) were collected and dialyzed first against dialysis buffer A (50 mM Tris HCl, pH 8.0, 1 mM TCEP, 1 mM EDTA, 300 mM KCl), then dialyzed twice against dialysis buffer B (50 mM Tris-HCl, pH 8.0, 150 mM KCl, 1 mM DTT, 3 mM EDTA). After dialysis, protein concentration was determined by Bradford assay using IgG as the standard and divided by a factor of 2.6 [18], and was stored at -80◦ C.

A = A0 + rQ ([Q] + [EPQ])

(1)

The concentrations of both UV/Vis detectable molecular species ([Q] and [EPQ]) at time t were computed from their initial concentrations at time zero by numerically solving an initial-value problem defined by a system of differential equations (S1)–(S12) (Supporting Information). The numerical solution algorithm was the Livermore Solver of ODE Systems (LSODE) [21, 22]. The absolute global truncation error tolerance was 10−14 µM; the relative global truncation error was 10−8 (eight significant digits). For further details regarding data handling and analysis see Supporting Information. 2.2.3. Steady-state initial rate kinetics Steady state kinetics experiments were performed by measuring initial velocities at varying concentrations of NAD+ , NADH, and A110, by monitoring the production of NADH by absorbance at 340 nm ( = 6.22 mM−1 cm−1 ) using Hitachi U2000, or Shimadzu UV-2600 spectrophotometer. All measurements were performed in the assay buffer (50 mM Tris, 150 mM KCl, 1 mM DTT, pH 8.0) at 25◦ C with saturating concentration of IMP (1 mM) and 14 nM enzyme (nominal concentration) in a total of 1 ml assay volume in 1 cm path length cuvettes. Initial rates were determined by either linear or exponential fit of the first five minutes of the assay (17 time points stepping by 20 seconds). Steady-state initial rate data, obtained while simultaneously varying the concentrations of NAD+ , NADH, and A110, were combined into a single multi-dimensional data set and subjected to global regression analysis [19]. The fitting model is represented by Eqn (2), in which square brackets with lower index zero represent total or analytic concentrations of reactants; v0 is the reaction rate observed in the absence of inhibitors; and Ki∗

2.2.2. Stopped-flow pre-steady state kinetics Stopped-flow experiments were performed at 25◦ C using Applied Photophysics SX17MV spectrophotometer. Syringe 1 contained BaIMPDH∆L protein (nominal concentration 8 µM subsequently optimized during data analysis) and IMP (2 mM) in assay buffer containing 2.5% (v/v) DMSO. Syringe 2 contained variable concentrations of NAD+ (0.5, 1, 2, 4, 8, 12, and 16 mM) in assay buffer containing 2.5% (v/v) DMSO. Syringe 2 also optionally contained either 120 µM NADH, as product inhibitor, or 12 µM A110, as inhibitor of interest. After preincubation, the contents of both syringes were mixed in the 1:1 ratio and the reaction progress was monitored by recording NADH absorbance at 340 nm. Each co-injection resulted in 10,000 time-points spanning from t = 0 to t = 2.5 sec, stepping by ∆t = 0.25 msec. Each individual data trace (absorbance vs. time) for kinetic analysis was obtained as an average of 12 separate injections. The experiments with NAD+ alone were replicated four times, on separate days, starting from fresh stock solutions in each daily session. The experiments with added NADH were 2

is the apparent inhibition constant corresponding to the given kinetic mechanism; see Eqns (S25)–(S33), (S41) and (S42) for definitions of both v0 and Ki∗ corresponding to the kinetic mechanism in Scheme 2. Note that at full IMP saturation, [E]0 stands for the total concentration of E·IMP complex.

v = v0

[E]0 − [I]0 − Ki∗ +

1 and Supplementary Figure S2. Raw data files are available in the Supporting Information. 3.1.1. Minimal kinetic mechanism For the purposes of this report, a “minimal” kinetic mechanism is defined as one where all individual steps and the associated microscopic rate constants – or at least their limiting values ( the lower or upper bounds) – are fully determined by the available experimental data. In contrast, a “redundant” kinetic mechanism contains rate constants that are assumed to exist on the basis of external evidence but are not directly supported by experimental data under consideration. In this work we set out to identify the minimal kinetic mechanism for the available stopped-flow transient kinetic data and the best-fit values, or at least the lower or upper bounds, for all microscopic rate constants appearing in the minimal mechanism. The starting point for the development of a minimal kinetic model is the redundant mechanism displayed in Scheme 1. The naming scheme for substrates (A, B) and products (P, Q) follows IUB/IUPAC recommendations [36] as well as conventions commonly used in classic enzyme kinetic texts [37]. The dash in E-P represents the covalent intermediate. This reaction scheme is based on numerous previously published reports (for review cf. [8]). The covalent intermediate species E-P is proposed to exist in two distinct conformations (“open” and “closed”, [8]) which are assumed to interconvert essentially instantaneously on the time scale of the experiment. By using the trial-and-error approach, eliminating either one microscopic step at a time or groups of microscopic steps until all remaining rate constants were fully defined by the data, the redundant kinetic mechanism in Scheme 1 was reduced to the minimal kinetic mechanism in Scheme 2. The tilde symbol in the species name signifies that E˜ P could be either the covalent intermediate on the reaction pathway or the noncovalent enzyme–product complex. The stopped-flow transient kinetic experiment does not provide sufficient information to distinguish between the two scenarios. Under IMP saturating conditions, the concentration of free enzyme E is by definition zero and therefore grayed segment in upper left corner of Scheme 2 is not operational. Solid black arrows with rate constant names appended to them represent those microscopic rate constants that can be unambiguously determined from the stopped-flow transient kinetic data, not only in terms of their best-fit values, but also including both the lower and the upper bounds. The four dashed arrows represent those microscopic rate constants (k4 , k−4 , k7 , and k−7 ), for which only the lower limit can be determined but not the upper limit. Rate constants k4 and k−4 represents product inhibition by NADH, whereas rate constants k7 and k−7 represent substrate inhibition by NAD+ . Both steps can be characterized as taking place with instantaneous equilibration (rapid equilibrium approximation). The equilibrium dissociation constants for these two steps are well defined by the data. Figure 1 displays seven of 21 kinetic traces that were all analyzed as a single global unit [19]. The complete set of 21 traces is shown in Figure S2. In the particular experiment illustrated in Figure 1, NAD+ was varied in the presence of added A110.

q 2 [E]0 − [I]0 − Ki∗ + 4 [E]0 Ki∗ 2 [E]0

(2) For further details regarding data handling and analysis see Supporting Information. 2.3. Data Analysis Nonlinear least-squares regression analysis was performed by using a custom implementation of the Levenberg-Marquardt algorithm [23, 24]. For verification, all regression analyses were repeated by using the hybrid Trust-Region data fitting method [25–27] (algorithm NL2SOL, version 2.3) with usersupplied Jacobian matrix of first derivatives. Initial estimates of microscopic rate constants were discovered with the aid the Differential Evolution global minimization algorithm [24, 28]. For further details regarding initial estimates see Supporting Information, section 2.2, pp. 7–8. Asymmetric confidence interval for adjustable regression parameters (kinetic constants, initial concentrations, and offset on the signal axis) were determined by using the profile-t search method of Bates & Watts [29–31]. In the case of steadystate initial rates, the confidence level for marginal confidence intervals of model parameters (as opposed to joint confidence regions, which were not evaluated) was 95%. However, in the case of stopped-flow pre-steady state kinetics data, the individual times points are not statistically independent. Therefore the critical value of the residual sum of squares was chosen by using the empirical approach advocated by Johnson [32–34]. According to this method, the parameter space is searched until the best-fit residual sum of squares increases by a reasonably large percentage of its best-fit value. All analyses reported here used ∆SSQ = 5%. All data analyses were performed by using the software package DynaFit [24, 35]. The DynaFit input script files (model specification, initial values of nonlinear regression parameters, and the method of analysis) are listed in full in the Supporting Information. 3. Results and Discussion 3.1. Stopped-flow kinetics We performed eight replicated series of stopped-flow experiments in four separate daily sessions. We used a CBS subdomain deleted variant of BaIMPDH∆L (the same as the crystal structure [15]). Enzyme (nominal concentration 4 µM) was preincubated with fixed saturating concentrations of IMP (1 mM; Km = 70 µM) and mixed with varied concentrations of NAD+ (0.25 – 8 mM) in the presence or absence of NADH (60 µM) or A110 (6 µM). Representative data are shown in Figure 3

E-P.B E.B

k1'

k7 k-7

+A

+B

k-2'

k-1'

k2' k3

E

E.A.B

+A

k-1

+B

k2 k1

k5

k4 E-P

E-P.Q

k-2 +I

k-8

k8

k-9 A B P Q I

IMP NAD+ XMP NADH inhibitor

+I

k9

k-10

k10

E.P.I

E-P.I

E +P k-6

+I

+B

E.A

H2O

+Q k-4

k-3

k6 E.P

+I

k-11 k11 E.I

E.A.I

Scheme 1

Each individual experiment involves multiple distinct phases, as evidenced by the presence of two distinct “shoulders” visible in the kinetic trace associated with the [NAD+ ] = 8 mM. The corresponding instantaneous rate plots (Figure S2, lower right panel) further support the observed multi-phasic nature of the transient data. Note that the enzyme and inhibitor concentrations are comparable in magnitude (“tight binding” [38]), which means that simplified pseudo-first-order approximation does not hold and therefore it would not be theoretically justified to analyze the transient kinetic data by the conventional multi-exponential analysis. Instead, one must indeed resort to a global mathematical model formulated as a system of simultaneous differential equations. Figure 2 shows the evolution of enzyme species concentrations corresponding to the [NAD+ ] = 1 mM kinetic trace displayed in Figure 1. Note again the highly complex shapes of the plots, essentially defying any possible use of the conventional multi-exponential analysis, as the overall rate of product formation is proportional to the instantaneous concentration of E·A·B (blue dashed curve). The two enzyme–inhibitor complexes are formed on different time scales and with different abundance at steady-state. The enzyme–substrate–inhibitor complex E·A·I is dominant during the pre-steady state phase of the experiment up to approximately t = 100 msec. The enzyme–product–inhibitor complex E˜ P·I is strongly dominant at steady state (t > 1 sec), although some amount of E·A·I also persists. This result predicts that in steady-state initial rate measurements performed under identical experimental conditions (i.e., at saturating concentrations of IMP), A110 should be identified as a mixed predominantly uncompetitive inhibitor. Based on the concentration plot in Figure 2, the uncompetitive inhibition constant can be predicted to be significantly smaller than its competitive inhibition constant.

from 16 independent combinatorial replicates (see Supporting Information for details regarding combinatorial replication). Table 1 list the averages (n = 16) and the associated standard deviations from replicates for the best-fit values of rate constants and also for the corresponding lower and upper bounds evaluated by the 5% ∆SSQ according to Johnson’s empirical method [32–34]. Similar results (not shown) were obtained at the more stringent 10% ∆SSQ confidence level. The microscopic rate constants listed in Table 1 fall into four categories according to how well they are determined by the experimental data. In the first category are three of the four substrate catalytic constants pointing in the forward directions (k2 , k3 , and k5 ) and also the dissociation rate constants for both inhibitor binding steps (k−8 and k−9 ). All five rate constants listed above are very well defined by the available data, as the coefficient of variation from replicates (n = 16) is lower than 5% in all cases. In the second category are the reverse catalytic rate constants (k−2 and k−3 ) and the inhibitor association rate constants (k8 and k9 ). These four constants are marginally less well determined, as the corresponding coefficient of variation is approximately between 10% and 25%. However, both the upper limit and the lower limit of the asymmetric confidence interval is well defined for all four rate constants. In the third category are the two microscopic rate constants that characterize the substrate inhibition step, k7 and k−7 . Both best-fit values are well defined, with CV < 20%. The lower limit of the confidence intervals for k7 and k−7 is also very well defined, with CV < 25%. The upper limit of the confidence interval is not defined at the 5% ∆SSQ confidence level, which means that the range of plausible values for k7 and k−7 spans from their corresponding lower limits essentially to infinity. However, all pairs of k7 and k−7 values that were located within the 5% ∆SSQ empirical confidence intervals maintained a nearly invariant ratio, k−7 /k7 = 5.5 mM (Figure S5). Thus the

3.1.2. Rate constant bounds To establish the bounds on rate constants appearing in Scheme 2, the results of nonlinear regression analysis were averaged 4

A B P Q I

E.B

IMP NAD+ XMP NADH inhibitor

E~P.B k7 k-7 +B

k3 E

k4

E.A.B k2

E~P

E-P.Q +Q k-4

k-3

+I

+B

k-2 E.A

k-9 k5

E~P.I

+I

k-8

k8

k9

+A saturating

P

E.A.I

Scheme 2

equilibrium dissociation constant associated with NAD+ substrate inhibition is well defined by the transient kinetic data. Finally in the fourth category are the two rate constants that characterize product inhibition by NADH, k4 and k−4 . This is the only pair of microscopic rate constants where the best-fit values are poorly defined by the data, as the coefficient of variation is greater than 100% in both cases. However the ratio of both rate constants, i.e. the corresponding dissociation equilibrium constant, remains nearly invariant across all 16 replicated measurements and it is approximately equal to Kd(EP.NH) = 100 µM. The upper limits at the 5% ∆SSQ confidence level are undefined. Only the lower limits of both k4 and k−4 are well defined, with CV < 25% (Table S4). In particular the lower limit of the NADH association rate constant is approximately k−4 > 2 µM−1 s−1 . All pairs of k4 and k−4 values that were located within the 5% ∆SSQ empirical confidence intervals maintained a nearly invariant ratio, k4 /k−4 = 100 µM (Figure S5). Thus the equilibrium dissociation constant associated with NADH product inhibition is well defined by the pre-steady state kinetic data. Even though the upper limits of the bimolecular association rate constants k−4 and k7 are not sharply defined by the available transient kinetic data, those upper limits are imposed by physical constraints, in particular by diffusion control. Theoretical calculations predict that the “diffusion-controlled encounter frequency of an enzyme and a substrate should be about 109 M−1 s−1 ” [39, pp. 164-166]. The highest experimentally observed values [39] frequently fall in the range between 106 and 108 M−1 s−1 . The lower limit of k4 (NADH rebinding) is approximately 2 × 106 M−1 s−1 , which means that NADH rebinding is extremely rapid. In contrast, the lower limit of k−7 (substrate inhibition by NAD+ ) is much lower, approximately 4 × 103 M−1 s−1 . In summary, out of 13 microscopic rate constants that ap-

pear in Scheme 2, 11 rate constants (i.e., all except k4 and k−4 ) were determined uniquely in terms of their well-reproduced best-fit values. Additionally, nine rate constants have well defined upper and the lower limits at the 5% ∆SSQ confidence level, according to Johnson’s empirical method [33]. Johnson’s ∆SSQ method certainly represents a massive improvement over the conventional and frequently meaningless “standard error” method of assessing the uncertainly of nonlinear model parameters [40, pp. 696-698, Eqn (15.6.4)]. However, it should also be noted that the empirical ∆SSQ method has no basis in rigorous statistical theory [29, 41–43]. Instead, the investigator must choose an arbitrary “threshold” [33] value for ∆SSQ (for example, 5%, 10%, or 25%), based entirely on subjective personal preferences. Thus, it is possible that the distinct minima clearly visible in the likelihood profiles [44] for k7 and k−7 (see Fig. S4, lower panels) do in fact represent meaningful best-fit values. Importantly, the likelihood profiles k4 and k−4 (see Fig. S4, upper panels) are perfectly flat, without even the slightest hint of a true minimum on the least-squares hypersurface. Never-the-less, to remain firmly on the safe side, in the analysis of steady-state initial rates (see below), we have chosen to treat with full confidence only the nine rate constants that have clearly defined upper and lower limits at 5% ∆SSQ. It is highly unusual that at least nine microscopic rate constants should be uniquely determined from a single globally analyzed transient kinetic data set. In fact, we are not aware of any published report to that effect, although a large number of microscopic rate constants have been previously determined from multiple independent experiments, analyzed separately. In this fashion, Benkovic et al. [45, 46] determined 22 microscopic rate constants in the catalytic mechanisms of dihydrofolate reductase. Similarly, Anderson et al. [47] determined 12 microscopic rate constants in 8 separate rapid quench kinetic experiments (see Figs. 1 – 8 in ref. [47]), each of which was 5

[NADH] = 0, [A110] = 6 µM

20 0

∆A340, mOD

40

0.25 0.5 1 2 4 6 8 mM

residuals

0.5 0 -0.5 -1 -1.5

0.01

0.1

1

time, s

Figure 1: A representative data set from the global fit of stopped-flow transient kinetic data to the kinetic mechanism in Scheme 1. Concentrations: [IMPDH] = 4.0 µM (nominal), [IMP] = 1.0 mM, [A110] = 6 µM (nominal), [NAD+ ] see labels in Figure margin. values were kon = (10, 20, 50, 100, 1000) µM−1 s−1 . The purpose was to determine to what extent (if any) the best-fit values of the remaining rate constants appearing in Scheme 2 are affected by the arbitrary choice of kon = k−4 = k7 . The results are summarized in Table S5. Within the examined range of the kon values, the best-fit values of the corresponding koff rate constants were such that the Kd = koff /kon remained invariant. Specifically for product inhibition by NADH, the dissociation equilibrium constant k4 /k−4 remained within 5% of 100 µM. Similarly for substrate inhibition by NAD+ , the dissociation equilibrium constant k−7 /k7 remained within 10% of 5.5 mM. Importantly, the remaining microscopic rate constants appearing in Scheme 2 varied by less than 5% in response to arbitrary variations in the assumed kon value spanning three orders of magnitude. Thus the ultimate values of microscopic rate constants that were subjected to subsequent validation by steady-state initial rate kinetic measurements are those that are listed in the right-most column of Table S5. These values assume that the rapid-equilibrium assumption is sufficiently well characterized by kon = 100 µM−1 s−1 .

focused on a particular sub-set of the overall twelve-step mechanism. The largest number of microscopic rate constants ever determined in a global fit of a single global data set is six, as reported by Schroeder at al. [48, 49]. The challenges of successful model identification grow very much faster than linearly with an increase in the number of adjustable model parameters. Thus, a kinetic mechanism containing either nine or even 11 adjustable rate constants is significantly more difficult to establish by fitting a single data set, compared to a kinetic mechanism with six microscopic rate constants – the largest number reported so far [48, 49]. Therefore, in order to stress-test our regression model in terms of reproducibility, we have performed 16 replicated regression analyses, by utilizing the combinatorial “mix-and-match” method described above. The successful determination of at least nine microscopic rate constants in the global analysis [19] of a single combined transient kinetic data set was almost certainly facilitated by the fact that the UV/Vis spectrophotometric signal was sensitive not only to the presence of the final reaction product (NADH, Q in Scheme 2), but very importantly also to the presence of the enzyme–product complex (E-P.Q in Scheme 2; see also Eqn (1)).

3.2. Steady-state initial rate kinetics 3.2.1. Predicted vs. observed reaction rates Steady-state initial rates were determined in four types of experiments, depending on the varied reaction component(s). In all these experiments, the substrate concentration was held fixed at the saturating concentration [IMP] = 1.0 mM. The fact that [IMP] = 1.0 mM is indeed saturating can be established by inspection of Figure S9, upper left panel. Note that two IMP saturation curves observed at [IMP] = 0.75 mM and 1.0 mM, respectively, are virtually indistinguishable.

3.1.3. Partially constrained model Based on the fact that only lower limits can be determined for kinetic constants that characterize either substrate inhibition by NADH (k4 and k−4 ) or product inhibition by NAD+ (k7 and k−7 ), the kinetic model shown in Scheme 2 was stress-tested in a series of regression analyses where the association rate constants k−4 and k7 were both assigned arbitrarily chosen “rapid equilibrium” values. The five arbitrarily chosen rate constant 6

[NADH] = 0, [A110] = 6 µM, [NAD] = 1 mM

1 0

concentration,

µM

2

E.A E.A.B E~P.Q E~P E~P.B E.A.I E~P.I

0.01

0.1

1

time, s

Figure 2: The evolution of enzyme species concentrations corresponding to the [NAD+ ] = 1 mM kinetic trace displayed in Figure 1. The dominant enzyme–inhibitor complex up to approximately t = 100 msec is E·A·I. However, at steady state (t > 1 sec) the dominant enzyme–inhibitor complex is E˜ P·I. In the first type of experiment, aimed at establishing the cofactor saturation curve, [NAD+ ] was varied between 0.1 mM and 8.0 mM, in the absence of either NADH as product inhibitor or A110 as the inhibitor of interest. This experiment was performed in triplicate. In the second type of experiment, aimed at the product inhibition effect of NADH, [NAD+ ] was varied between 0.1 mM and 1.0 mM at various levels of [NADH] (0, 50, 100, and 150 µM). In the third type of experiment, aimed at the inhibition properties of A110, [NAD+ ] was varied between 0.12 mM and 1.2 mM at various levels of [A110] (0, 15, 30, 60, 120, and 180 nM). The fourth and final type of experiment was a variation on the immediately preceding experiment type and involved relatively high concentrations of [NAD+ ], varied between 1.0 mM and 8.0 mM at various levels of [A110] (0, 15, 30, 60, 90, and 150 nM). The experimental data from all four types of experiment were combined into a single super-set of data and analyzed by the global fit [19] method. The nonlinear regression model was Eqn (2). The various algebraic terms that define the uninhibited rate v0 , Eqn (S41), and the apparent inhibition constant Ki∗ , Eqn (S42), are defined as shown in Eqns (S25)–(S33). In Eqns (S25)–(S33), the assumed values of all microscopic rate constant were those obtained in the stopped-flow transient kinetic study, namely: k2 = 0.0318 µM−1 s−1 ; k−2 = 10.9 s−1 ; k3 = 82.1 s−1 ; k−3 = 44.3 s−1 ; k4 = 11500 s−1 ; k−4 = 100 µM−1 s−1 (rapid equilibrium approximation); k5 = 13.6 s−1 ; k7 = 100 µM−1 s−1 (rapid equilibrium approximation); k−7 = 566000 s−1 , k8 = 13.9 µM−1 s−1 ; k−8 = 11.0 s−1 ; k9 = 5.51 µM−1 s−1 ; and k−9 = 0.27 s−1 . Importantly, in this first round of initial rate analysis, all microscopic rate constants listed above were held fixed at values derived from the stopped-flow experiment. The only adjustable model parameter in the regression Eqn (2) was the active enzyme concentration, [E]0 . The enzyme concentration was op-

timized locally for each particular type of experiment, because the different steady-state experiments were performed over the period of approximately one year and thus the active enzyme concentration might have changed slightly over time. The nominal enzyme concentration was 14 nM, and the best fit values of the active-site concentration varied from 8.4 nM to 10.7 nM. Given the fact that all rate constants were held fixed in the regression and only the enzyme concentration was optimized, the highly restricted “fit” of the combined initial rate data is merely a comparison between the observed initial rates and those that are predicted by the theoretical model derived form the stopped-experiment. The results are shown in Figure 3. The full text of the requisite DynaFit input file is listed in Appendix SA.2.2; the best-fit values of [E]0 are listed in Table S6. Note that that throughout this report the reaction rates are expressed in directly observable absorbance units per second, rather than in concentration units. The main reason for this choice is that, in particular in the analysis of stopped-flow data, the molar response coefficients are generally treated as adjustable model parameter. In the specific case of Figure 3, the observed reaction rate (approximately 0.4 × 10−3 dimensionless absorbance units per second) corresponds to the chemical reaction rate of approximately 0.4/6.22 = 0.064 µM/sec in NADH formation. The results displayed graphically in Figure 3 show that the 13 microscopic rate constants determined by the stopped-flow transient kinetic measurements predict the steady-state initial rate data reasonably well. For example, the position of the maximum on the NAD+ saturation curve (approximately at [NAD+ ] = 1.5 mM), in the upper left panel, is well predicted, as is the slope of the downward portion due to substrate inhibition. In the upper right panel of Figure 3, there is a good agreement between the predicted (curves) and observed (symbols) product inhibition effect of NADH. The lower left panel shows the predicted vs. observed inhibition effect of A110 at relatively 7

unit k2 k−2 k3 k−3 k4 k−4 k5 k7 k−7 k8 k−8 k9 k−9

−1 −1

µM s s−1 s−1 s−1 s−1 µM−1 s−1 s−1 µM−1 s−1 s−1 µM−1 s−1 s−1 µM−1 s−1 s−1

best-fit 0.034 13 91 37 360000 3500 14.4 0.008 44 12 12 4.1 0.264

± ± ± ± ± ± ± ± ± ± ± ± ±

0.001 3 3 10 400000(b) 4000(b) 0.4 0.002 7 0.9 0.5 0.5 0.004

lower limit 0.03 4.9 81 18 180 1.7 13.6 0.0044 24 8 9.7 2.6 0.213

± ± ± ± ± ± ± ± ± ± ± ± ±

0.001 2 3 6 40 0.4 0.5 0.0006 2 0.9 0.4 0.4 0.004

upper limit 0.04 32 110 86

± ± ± ±

0.002 7 8 40

(a) (a)

15 ± 0.4 (a) (a)

17 16 6 0.321

± ± ± ±

1 0.6 0.6 0.006

(a)

Upper limit is undefined at the 5% ∆SSQ confidence level [33]. However the dissociation equilibrium constants k4 /k−4 and k7 /k−7 were invariant during the confidence interval search, see Figure S5.

(b)

Coefficient of variation (%CV) greater than 100%. However the dissociation equilibrium constant k4 /k−4 is very well defined by the data across all 16 combinatorial replicates, see Figure S6.

Table 1: Averages (n = 16) and the corresponding standard deviations of microscopic rate constants from replicates determined by the global fit of stopped-flow transient kinetic data. The lower and upper limits are asymmetric confidence intervals determined according to the method proposed by Johnson [32–34] at the 5% ∆SSQ level. See also Figures S4–S6, Supporting Information. low concentrations of NAD+ . Again the relative spacing of the substrate saturation curves is well described by the theoretical model derived from the stopped-flow data. The same applies to the lower right-hand panel of Figure 3, displaying the inhibitory effect of A110 at relatively high concentrations of NAD+ . 3.2.2. Predicted vs. observed composite kinetic constant Based on the best-fit values of microscopic rate constants determined in the stopped-transient kinetic experiments (see Table 1) and given the definition of steady-state kinetic constants as shown in Eqns (S25)–(S33), the predicted values of kinetic constants are listed in Table 2, column “predicted”. In order to validate this prediction, the combined initial rate data was again fit globally [19] to Eqn (2), this time while treating all kinetic constants (kcat , Km(B) , Ki(I) , Ki(I,B) , Ki(B) , Ki(Q) , Ki(Q,B) ) as adjustable model parameters. The best-fit values obtained in this second round fitting the initial rate data are shown in numerically in Table 2 and graphically in Figure S9. The results of fit in good agreement with the values predicted from the theoretical model derived the stoppedflow transient kinetic experiment. The “uncompetitive” inhibition constant Ki(I,B) predicted from stopped-flow measurements is 57 nM; the experimentally observed value from initial rate measurements was (51 ± 2) nM, with the 95% confidence level interval spanning from 47 to 54 nM. In previous reports on the kinetics of full-length BaIMPDH, the “uncompetitive” inhibition constant for A110 with respect to variable NAD+ was reported as (58 ± 4) nM [13] and (57 ± 7) nM [50]. The IC50 of A110 inhibiting BaIMPDH∆L was reported as (43 ± 3) nM at 1.0 mM IMP and 1.5 mM NAD+ . 8

The “competitive” inhibition constant Ki(I) predicted from stopped-flow measurements is 0.8 µM; the experimentally observed value from initial rate measurements is (0.5 ± 0.2) µM, with the 95% confidence level interval spanning from 0.3 to 1.5 µM. Thus the initial rate experiments confirm the prediction of inhibition mode based on the results of the stopped-flow experiments. A110 was predicted to behave as a mixed predominantly uncompetitive inhibitor of BaIMPDH∆L and this is in fact confirmed by initial rate measurements. 3.2.3. Predicted vs. observed kinetic isotope effect The stopped-flow kinetic results yielded distinctly nonzero best-fit value of the E·A·B → E·A + B dissociation rate constant, k−2 , see Table 1. However, k−2 appears to be effectively zero (i.e., NAD+ is a sticky substrate) for at least some IMPDHs [51]. To verify the stopped-flow kinetic result in this respect, the D (kcat /Km ) isotope effect was determined by assaying BaIMPDH∆L with saturating 1 H-IMP and 2 D-IMP. The results are reported in detail in Supporting Information, section 3.3.4. (Table S7). Briefly, the value of k−2 listed in Table 1 predicts that D (kcat /Km ) should be significantly greater than one. On the other hand if k−2 were negligibly small (“sticky” substrate) then the predicted value of D (kcat /Km ) should be by definition equal to one. The observed value is (2.4 ± 0.1). The isotope effect on kcat /Km(B) confirms that that NAD+ dissociates readily from the E·A·B complex, in agreement with the pre-steady state kinetic results.

NAD

NAD + NADH 0.4

Q = 0 mM Q = 0.05 Q = 0.10 Q = 0.15

0

0

0.1

0.2

v, mOD/sec

0.2

v, mOD/sec

0.3

0.4

A = 0.75 mM A = 1.0 (a) A = 1.0 (b) A = 1.0 (c)

0

2000

4000

6000

[NAD+], µM

0

500

[NAD+], µM

(a)

(b)

NAD (low) + A110

NAD (high) + A110

0.3

0.4

I = 0 nM I = 15 I = 30 I = 60 I = 90 I = 150

0

0

0.1

0.2

v, mOD/sec

0.4

I = 0 nM I = 15 I = 30 I = 60 I = 120 I = 180

0.2

v, mOD/sec

1000

0

500

[NAD+], µM

1000

0

(c)

2000

4000

[NAD+], µM

6000

(d)

Figure 3: Comparison between observed initial rates (symbols) and initial rates predicted from the theoretical model derived for the stopped-flow transient kinetic data (curves). For details see text. Ki∗ with [P] were subjected to weighted linear regression analysis. The results of linear fit are shown graphically in Figure 4B. The dimensionless slope of the linear regression line was (7.0 ± 0.6) × 10−5 ; the intercept representing Ki∗ at [P] = 0 was (137 ± 6) nM. Figure S14 shows the result of a heuristic simulation designed to allow an interpretation of the experimentally observed dependence. The simulation results can be summarized as follows. If the binding affinity of A110 toward the noncovalent intermediate E·P were greater than the binding affinity toward the covalent complex E-P, the plot of Ki∗ vs. [P] would be sloping downward. If both binding affinities were nearly identical, the plot would be approximately horizontal. Finally, if the binding affinity of A110 toward the covalent intermediate E-P were dominant, the plot of Ki∗ vs. [P] would be sloping upward.

3.2.4. A110 plus XMP double-inhibitor experiment To determine if A110 binds to E-P or E·P, the apparent inhibition constant for A110 was determined at varying concentrations of [XMP] (0.125 – 2.0 mM) in the presence of subsaturating concentrations of IMP (0.2 mM) and NAD+ (1.5 mM). Experimental dose-response curves were fitted to Eqn (2), where [E]0 was held fixed at 10 nM while v0 and Ki∗ were treated as optimized model parameters. The raw experimental data are shown in Figure 4A. The best-fit values of the apparent inhibition constant at various XMP concentrations and the associated formal standard errors from nonlinear regression are listed in Table S8. The apparent inhibition constant increased approximately linearly with an increase in the product concentration (Figure 4B). This observation suggests that E·XMP does not form a high affinity complex of A110. The observed changes in 9

#

parameter −1

1 2 3 4 5 6 7

kcat , s Km(B) , µM Ki(I) , µM Ki(I,B) , µM Ki(B) , µM Ki(Q) , µM Ki(Q,B) , µM

predicted

observed ± std.err.

11.6 415 0.791 0.0572 6610 301 87.2

11.6 460 0.5 0.051 7400 620 97

± ± ± ± ± ± ±

0.2 20 0.2 0.002 400 360 9

cv,%

low

high

1.8 4.0 43.6 4.3 5.3 57.7 8.9

11.2 430 0.3 0.047 6800 320 84

12.0 490 1.5 0.054 8100 4600 113

Table 2: Comparison of predicted and observed steady-state kinetic constants from global fit of combined initial rate data to Eqn (2), where kinetic constants are defined by Eqns (S25)–(S33). The columns labeled “low” and “high” are lower and upper limits of the asymmetric confidence interval [29, 31] at 90% likelihood level. For further details see text. A110 : Ki

(app)

A110 : Ki(app) vs. [XMP]

vs. [XMP]

0.3

0.8

P = 0 uM P = 125 P = 250 P = 500 P = 1000

0

0

0.2

0.1

0.2

Ki(app), µM

0.4

v, mOD/sec

0.6

P = 2000

0

0.1

0

0.2

[I], µM

1000

[XMP], µM

(a)

2000

(b)

Figure 4: A110 plus XMP double-inhibitor experiment. (a) Determination of the apparent inhibition constant for A110 at various fixed concentrations of [XMP]. (b) Linear least squares fit of experimentally observed values of Ki∗ vs. [P]. The experimentally observed plot has an upward slope, signifying that A110 does not bind predominantly to the noncovalent product complex but instead binds preferentially to the covalent intermediate. The mathematical details are are explained in Supporting Information, section 3.3.5. A very approximate estimate of the ratio of the two relevant inhibition constants suggests that the binding to E·P might be at least an order of magnitude weaker than the binding to E-P. 3.3. Conclusions IMPDH controls the guanine nucleotide pool, and thus proliferation, in virtually every organism. Human IMPDH inhibitors are used as immunosuppressive, antiviral and anticancer therapy, and microbial IMPDHs have emerged as potential drug targets. Prokaryotic and eukaryotic IMPDHs bind NAD+ in distinctive sites that recognize very different cofactor conformations [15]. This difference has been exploited to develop selective inhibitors of CpIMPDH in six different frameworks. The cofactor binding site of CpIMPDH is very similar to that found IMPDHs from from many pathogenic bacteria, including 10

B. anthracis, Campylobacter jejuni, C. perfringes, Streptococus pyogenes, Heliobacter pylori and M. tuberculosis [5]. Despite this similarity, the affinity of the inhibitors for these enzymes can vary by 100-1000-fold. Structures have been solved of several inhibitors bound to E·IMP complexes, but these structures do not reveal the basis for this surprising variation. Here we have delineated the kinetic mechanism of one representative inhibitor, A110, as shown in Scheme 3, which displays the dominant inhibitor binding mode. According to Scheme 3, this compound binds preferentially to the covalent intermediate. Thus the crystal structures do not represent the high affinity enzyme-inhibitor complex, which may explain why they do not provide insight into the varied spectrum of IMPDH inhibitors. Acknowledgements The authors thank Cynthia Tung for assistance with the synthesis of A110. This work was supported by National Institutes of Health grants AI093459 and GM054403 (to L. H.).

IMP O NAD+

N

HN H

N

N

HN

N R

Cys S

Cys-S

XMP O

E-XMP* O

NADH

N

H2O

N R

O

E

N

N R

Cys SH

E

E

A110

O

Cys S

O N

O

N

HN

E

N

HN

N

N R

N N N

• Cl

E-XMP* • A110

Scheme 3

Supporting Information Available

[6] L. Hedstrom, G. Liechti, J. B. Goldberg, D. R. Gollapalli, The antibiotic potential of prokaryotic IMP dehydrogenase inhibitors, Curr. Med. Chem. 18 (2011) 1909–1918.

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13

Inhibition of IMPDH from Bacillus anthracis. Mechanism revealed by pre-steady state kinetics S UPPLEMENTARY I NFORMATION

Yang Weia,1 , Petr Kuzmiˇcb , Runhan Yua , Gyan Modia , Lizbeth Hedstrom∗,a a Departments

of Biology and Chemistry, Brandeis University, Waltham, Massachusetts b BioKin Ltd., Watertown, Massachusetts

Abstract This Supplementary Information document contains the detailed description of (a) the raw experimental data; (b) the mathematical models; (c) the nonlinear regression procedures. This information relates to the analysis of both the stopped-flow transient kinetic data and the steadystate initial rate data on the inhibition of IMPDH from Bacillus anthracis by the small-molecule inhibitor A110 specifically under experimental conditions where IMP as substrate is present at saturating concentrations. This document also contains input files for the software package DynaFit (Kuzmic, 1996, 2009) which was used to perform all kinetic analyses. Key words: enzyme kinetics; inhibition; IMP dehydrogenase; B. anthracis; mathematics; statistics

Contents 1 Introduction 2 Stopped-flow transient kinetics 2.1 Experimental data . . . . . . . . . . . . . . . . . . . . 2.2 Theoretical model . . . . . . . . . . . . . . . . . . . . 2.3 Stopped-flow kinetic results . . . . . . . . . . . . . . . 2.3.1 Representative global data set . . . . . . . . . 2.3.2 Combinatorial replicates . . . . . . . . . . . . 2.3.3 Partially constrained model: Rapid equilibrium 2.4 Pre-steady state kinetics: Summary . . . . . . . . . . .

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author address: Sanford-Burnham Institute, La Jolla, California

Biochemistry Ms. No bi-2016-00265u.R2

Supporting Information

3.3

3.4

3.2.1 “Classical” rate equation (King-Altman & Cleland) 3.2.2 “Tight binding” rate equation (Morrison) . . . . . Steady-state initial rate results . . . . . . . . . . . . . . . 3.3.1 Predicted vs. observed initial rates . . . . . . . . . 3.3.2 Predicted vs. observed kinetic constants . . . . . . 3.3.3 Diagnostic Eadie-Hofstee plots . . . . . . . . . . . 3.3.4 Predicted vs. observed H/D isotope effect . . . . . 3.3.5 Double inhibition experiment . . . . . . . . . . . Steady-state kinetics: Summary and conclusions . . . . . .

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4 Responses to Reviewers 39 4.1 Linear vs. logarithmic plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References

41

Appendix

44

A DynaFit scripts A.1 Global fit of stopped-flow data . . . . . . . . . . . . . . . . . . . . A.2 Global fit of initial rates . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Predict kinetic constants: “Classical” rate equation . . . . . A.2.2 Predict kinetic constants: “Tight binding” rate equation . . . A.2.3 Kinetic isotope effect . . . . . . . . . . . . . . . . . . . . . A.2.4 Apparent Michaelis constant of IMP . . . . . . . . . . . . . A.2.5 Apparent inhibition constant of XMP . . . . . . . . . . . . A.2.6 Apparent inhibition constant A110 in dependence on [XMP] A.2.7 Simulate dependence of Ki ∗ for A110 on [XMP] . . . . . . A.2.8 Linear fit of simulated Ki ∗ vs. [XMP] values . . . . . . . . A.2.9 Linear fit of experimental Ki ∗ vs. [XMP] values . . . . . . .

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B Raw data files and DynaFit script files B.1 Installation instructions . . . . . . . B.2 Organization of files and directories B.2.1 DynaFit script files . . . . . B.2.2 Raw experimental data files

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. . . .

. . . .

1. Introduction The main purpose of this Supporting Information document is to describe in sufficient detail the statistical and mathematical methods that were used for the analysis of (a) the stopped-flow transient kinetic data and (b) the initial rate kinetic data related to the inhibition of IMPDH 2 from Bacillus anthracis by the small-molecule inhibitor A110. The experimental setup includes simultaneous variations in the concentration three component species: 1. the cofactor, NAD+ ; 2. the coproduct, NADH; and 3. the inhibitor A110. All raw experimental data are attached to this report as a separate ZIP archive file, along with the relevant input files for that software package DynaFit [1, 2] that was exclusively used for data analysis. 2. Stopped-flow transient kinetics 2.1. Experimental data We performed eight replicated series of experiments ([NAD+ ] = 0.25, 0.5, ..., 8 mM) in four separate daily sessions. The experimental series where [NAD+ ] was varied alone, in the absence of NADH or A110, was replicated four times. In the discussion that follows as well, as in the attached raw data files, these four series of experiments are designated as replicates “N1” – “N4” . The series where [NAD+ ] was varied in the presence of NADH was replicated twice (replicates “H1” and “H2”). The series where [NAD+ ] was varied in the presence of A110 was also replicated twice (replicates “I1” and “I2”). To assure proper randomization. and thus increase the reliability of the regression analysis, groups of two replicated series were purposely obtained on different days, always starting from fresh stock solutions. The organization of the stopped-flow experiments is summarized in Table S1. The series replicate labels (N1, I1, etc.) correspond to CSV raw data file names (N1.CSV, I1.CSV, etc.) that are made available as part of this manuscript. session 1 2 3 4

replicate N1, I1 N2, I2 N3, H1 N4, H2

note

new batch of NAD+

Table S1: Organization of replicated stopped-flow experiments. Replicate labels correspond to available raw data file names. For details see text. 2 Abbreviations: DE, Differential Evolution algorithm; IMP, inosine 5’-monophosphate; IMPDH, IMP dehydrogenase; BaIMPDH, IMPDH from Bacillus anthracis; BaIMPDH∆L, BaIMPDH Glu92-Arg220 deletion mutant; MPA, mycophenolic acid; ODE, ordinary differential equations; SSQ, residual sum of squares; ∆SSQ, relative change in SSQ above the minimum value; XMP, xanthosine 5’-monophosphate

3

Each dataset for global analysis [3] consisted of 21 kinetic traces, namely, seven traces where [NAD] was varied in the absence of any additional component; seven traces where [NAD+ ] was varied in the presence of added NADH (product inhibition); and finally seven traces where [NAD+ ] was varied in the presence of the inhibitor A110. Given the number of available replicates, there exist 4 × 2 × 2 = 16 ways in which a group of 7 traces could be systematically drawn from the combined data set, namely: 4 ways to select a group of curves with [NAD+ ] only; 2 ways to select a group of curves with added NADH; and 2 independent ways to select a group of curves with added A110. In this work all 16 combinations of 21 traces were analyzed as statistically independent combinatorial replicates. The combinatorial replication scheme is summarized in Table S2. combinatorial replicate

“N” replicate

“H” replicate

“I” replicate

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

Table S2: Combinatorial replication scheme. Each of “N”, “H”, and “I” replicates consists of seven kinetic traces obtained at [NAD+ ] = 0.25, 0.5, 1, 2, 4, 6, 8 mM, either in the absence of any other component (“N”) or in the presence of [NADH] = 60 µM (“H”), or in the presence of [A110] = 6 µM (“I”). For details see text. All microscopic rate constants reported in this document, as well as the lower and upper limits of their respective confidence intervals, are simple averages and standard deviations determined from 16 (combinatorially) replicated global data sets as shown in Table S2. 2.2. Theoretical model The goal of our model-building effort was to identify a minimal mechanistic model for the available transient kinetic data. A minimal kinetic model is a reaction scheme where all microscopic rate constants (or at least their well defined limiting values) can be both structurally and practically identified [4] from the available experimental data. A hybrid modeling strategy combined (a) heuristic variations on previously published kinetic models [5, 6] and (b) global least-squares minimizations using the Differential Evolution (DE) algorithm [7]. For details regarding DE see [2, pp. 264–267]. The total evolutionary computing time was approximately 300 4

hours on a 2.4 GHz microprocessor (Intel-i7). The results are shown in Figure S1. Note that under conditions where the IMP substrate (S) is fully saturating ([IMP] = 1000 µM, >> Km(IMP) ) the free, uncomplexed enzyme species E effectively does not appear. A B P Q I

E.B

IMP NAD+ XMP NADH inhibitor

E~P.B k7 k-7 +B

k3 E

k4

E.A.B k2

E~P

E-P.Q +Q k-4

k-3

+I

+B

k-2 E.A

k-9

k5

E~P.I

+I

k-8

k8

+A saturating

k9

P

E.A.I

Figure S1: Minimal kinetic mechanism for the the inhibition BaIMPDH by A110. Dots represent noncovalent complexation; the dash character represents a covalent bond. The tilde character stands for either covalent bonding or noncovalent complexation, with the actual state being impossible to determine by stopped-flow kinetics alone.

Regression equation and model parameters The nonlinear least-squares regression model for individual kinetic traces is defined by Eqn (S1), where A is the absorbance (in mOD, 10−3 dimensionless absorbance units) at 340 nm at reaction time t > 0.004 sec; A0 is the adjustable baseline offset at t0 = 0.004 sec, essentially a property of the instrument; rQ = 6.22 mOD/µM is the molar response coefficient of NADH3 ; [Q] is the concentration of NADH at time t; and [EPQ] is the concentration of the ternary complex E-XMP∗ ·NADH. A = A0 + rQ ([Q] + [EPQ])

,

(S1)

The species concentrations [Q] and [EPQ] at time t were computed from their initial concentrations at time zero, by numerically solving an initial-value problem defined by the ODE system in Eqns (S2)–(S12). 3 Note that the molar response coefficient is not identical to the extinction coefficient of NADH, but rather it is 1000 times smaller. It is the number of dimensionless mOD units (10−3 dimensionless absorbance units) per micromole/liter of NADH generated in the reaction.

5

d[EA] dt

=

−k2 [EA][B] + k−2 [EAB] + k5 [EP] − k8 [EA][I] + k−8 [EAI]

(S2)

d[B] dt

=

−k2 [EA][B] + k−2 [EAB] − k7 [EP][B] + k−7 [EPB]

(S3)

d[EAB] dt

=

+k2 [EA][B] − k−2 [EAB] − k3 [EAB] + k−3 [EPQ]

(S4)

d[EPQ] dt

=

+k3 [EAB] − k−3 [EPQ] − k4 [EPQ] + k−4 [EP][Q]

(S5)

d[EP] dt

=

+k4 [EPQ] − k−4 [EP][Q] − k5 [EP] − k7 [EP][B] + k−7 [EPB] −k9 [EP][I] + k−9 [EPI]

(S6)

d[Q] dt

=

+k4 [EPQ] − k−4 [EP][Q]

(S7)

d[P] dt

=

+k5 [EP]

(S8)

d[EPB] dt

=

+k7 [EP][B] − k−7 [EPB]

(S9)

d[I] dt

=

−k8 [EA][I] + k−8 [EAI] − k9 [EP][I] + k−9 [EPI]

(S10)

d[EAI] dt

=

+k8 [EA][I] − k−8 [EAI]

(S11)

d[EPI] dt

=

+k9 [EP][I] − k−9 [EPI]

(S12)

The model equations listed above were automatically derived by the software package DynaFit [2] by using the coding listed below. The full text of the DynaFit script file is listed in Appendix A.1. In the code fragment below, the [mechanism] section is used within DynaFit to derive the ODE system Eqns (S2)–(S12). The [responses] section is used internally to derive the regression Eqn (S1). [mechanism] E.A + B E.A.B E.A.B E.P.Q E.P.Q E.P + Q E.P ---> E.A + P E.P + B E.P.B E.A + I E.A.I E.P + I E.P.I ... [responses]

: : : : : : :

k2 k3 k4 k5 k7 k8 k9

6

k-2 k-3 k-4 k-7 k-8 k-9

Q = 6.22 E.P.Q = 1 * Q Organization of model parameters Each global [3] data set, analyzed as a single unit, consisted of 21 kinetic traces of three different kinds as was described in section 2.1. Each kinetic trace was filtered down to 25 time points, such that the global data sets each contained 21 × 25 = 525 data points. The nonlinear least-squares regression model for each global set of 525 data points contained 36 adjustable parameters, some of which were “global” i.e. applicable to all 21 data traces; some of which were “local” to individual traces; and some of which were “semi-global” i.e. applicable jointly to a particular group of progress curves. This breakdown of regression parameters into three categories is explained in Table S3. parameter no.

parameter

type

applies to trace no.

1–13 14 15 16–36

k2 – k−9 initial [E.A]0 initial [I]0 offset A0

global global semi-global local

1–21, shared 1–21, shared 15–21, shared 1–21, individually

Table S3: Organization of regression parameters into categories. For details see text.

Initial estimates of model parameters In general, the nonlinear least-squares regression procedure requires that the investigator provides initial estimates of adjustable model parameters [8]. In unfavorable cases, the initial estimates must be surprisingly close the true values (which of course are not known at the outset), otherwise the least-squares minimization algorithm ends up in a false minimum. In preliminary data analyses (results not shown), the fitting model was found exquisitely vulnerable to falling into numerous false minima on the least-squares hypersurface. False convergence into a local least-squares minimum was reached frequently even when certain rate constants were shifted from their optimal value by less than a factor of 10. This local minimum problem was overcome by deploying the DE algorithm as a global least-squares minimizer [7]. The final set of nearly optimal initial estimates is specified in the DynaFit script listed in Appendix A.1: [constants] k2 = 0.01 ?? k3 = 100 ?? k4 = 1000 ?? k5 = 10 ?? k7 = 0.01 ?? k9 = 10 ?? k9 = 1 ??

, , ,

k-2 = 10 ?? k-3 = 10 ?? k-4 = 10 ??

, , ,

k-7 = 100 ?? k-8 = 10 ?? k-9 = 0.1 ??

Empirical confidence intervals The double question mark in the code fragment listed above signifies that the given rate constant are to be subjected to a systematic search to determine the asymmetric confidence interval 7

using the profile-t method of Bates & Watts [9–11]. However, rather than relying on the standard statistical formulas to determine the critical value of the residual sum of squares, we utilized the empirical method advocated by Johnson [12–14]. According to this empirical approach, the parameter space is searched until the best-fit residual sum of squares increases by a reasonably large percentage of its best-fit value. All analyses reported here used either ∆SSQ = 10% or ∆SSQ = 5%, as shown in Appendix A.1: [settings] {ConfidenceIntervals} SquaresIncreasePercent = 5 2.3. Stopped-flow kinetic results 2.3.1. Representative global data set Figure S2 4 shows the results of global fit of 21 combined kinetic traces, for one of 16 combinatorial replicates enumerated in Table S2 (in this case replicate N1H1I1, see listing in Appendix A.1). The corresponding best-fit vales of model parameters are shown in Table S4. The upper left panel shows 7 traces where NAD+ was varied alone; upper right panel shows 7 traces where NAD+ was varied in the presence of added [NADH] = 60 µM; the lower left panel contains 7 traces where NAD+ was varied in the presence of added [A110] = 6 µM (nominal). The lower right panel shows the plot of instantaneous reaction rates against reaction time. Note the complex shape of the instantaneous rate plot, in particular the lag phase between t = 0 and approximately t = 20 msec. Also note a “shoulder” feature between t = 0.1 sec and t = 0.5 sec. This complexity in the progress curve shape indicates that it is in principle impossible to apply the standard exponential or multi-exponential analysis of the reaction time course. In Table S4, most of the adjustable baseline offsets values were omitted for clarity. The “low” and “high” values were determined by the profile-t method [9–11] at ∆SSQ = 5% according to Johnson’s method [12–14]. The results listed in Table S4 show that 11 adjustable rate constants were determined with relatively low formal standard error, ranging from 4 to 17 percent (as coefficient of variation, cv). The only two exceptions are the rate constants k4 and k−4 , which characterize NADH dissociation and rebinding, respectively. Asymmetric confidence intervals The asymmetric confidence intervals for microscopic rate constants, as determined by the profile-t method [9] at the empirical 5% ∆SSQ level [14], can be grouped into three categories, depending on how well each rate constant can be identified from the available data. In the first category are all rate constants that appear in Figure S1 except those that characterize either product inhibition by NADH (k4 and k−4 ), or substrate inhibition by NAD+ (k7 and k−7 ). The likelihood profiles [4] for these nine rate constants (k2 , k−2 , k3 , k−3 , k5 , k8 , k−8 , k9 , and k−9 ) were all closed from both sides not only at the 5% ∆SSQ level, but also at the 10% and 25% ∆SSQ level (results not shown). A representative example profile is shown in Figure S3 for rate constants k2 and k−2 . The fact that the likelihood profiles in Figure S3 are closed from both sides means that k2 and k−2 are very well defined by the available pre-steady state kinetic data. 4 In order to alow multiple graphic panels to be displayed in the same illustration, all Figures in this document (EPS files generated by DynaFit [2]) were optimized for on-screen viewing, in the PDF format, rather than for printing, on paper. Please use the “View” ... “Zoom” ... “150%” in your PDF file viewer (e.g., Adobe Acrobat) in order to examine fine details.

8

[NADH] = 60 µM, [A110] = 0

[NADH] = 0, [A110] = 0

60 40

1000 ∆A340

0

0

1

residuals

0.5

residuals

0.25 0.5 1 2 4 6 8 mM

20

40 20

1000 ∆A340

60

80

0.25 0.5 1 2 4 6 8 mM

0 -0.5

0.01

0.5 0 -0.5

0.1

0.01

time, s

0.1

time, s

[NADH] = 0, [A110] = 6 µM

[NADH] = 0, [A110] = 6 µM

400

600

0.25 0.5 1 2 4 6 8 mM

200

d(1000 ∆A340) / dt

20 0

1000 ∆A340

40

0.25 0.5 1 2 4 6 8 mM

0 -1

0

residuals

0.5 -0.5 -1.5

0.01

0.1

1

0.01

time, s

0.1

1

time, s

Figure S2: Representative replicate (n = 16) from the least-squares fit. For details see text. In the second category are the two rate constants that characterize substrate inhibition by NAD+ (k7 and k−7 ). The likelihood profiles [4] for these two rate constants were closed from both sides at the 5% ∆SSQ level, but open from above at the 10% ∆SSQ level. This can be seen in Figure S4. Indeed at the upper end of the confidence intervals, the likelihood profile curves barely intersect the 5% ∆SSQ level and the profile become nearly horizontal at that level. In contrast, the lower end of the empirical confidence interval is very well defined. Johnson [12–14] advocates the use of 10% ∆SSQ as a stringent – albeit empirical – rule of thumb to assess the identifiability of a given rate constant. Thus in a highly conservative interpretation of the likelihood profiles displayed in Figure S4, the conclusion is that the upper limit are undefined for both k7 and k−7 . In other words, the data are consistent with the statement that k7 must be greater than 4.3 × 104 M−1 .s−1 (see Table S4) but there is no way of saying just high high k7 could actually be. Similarly, under the strict 10% ∆SSQ empirical rule, k−7 must be greater than 24 s−1 (see Table S4) but there is no way of saying just high high k−7 could 9

# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ... 36

param k2 k−2 k3 k−3 k4 k−4 k6 k7 k−8 k8 k−8 k9 k−9 [E.A]0 [I]0 offset / 1 offset / 2 ... offset / 21

unit −1

initial −1

µM .s s−1 s−1 s−1 s−1 µM−1 .s−1 s−1 µM−1 .s−1 s−1 µM−1 .s−1 s−1 µM−1 .s−1 s−1 µM µM mOD mOD ... mOD

0.01 10 100 10 1000 10 10 0.01 100 10 10 1 0.1 2.5 6 0 1 ... 12

final ± std.err. 0.0348 16 88.3 23.1 63000 700 14.39 0.0078 43.8 12.33 11.75 4.4 0.26 2.429 4.78 -0.118 1.213 ... 9.864

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.001 2.8 2.4 3.4 > 106 > 105 0.19 0.0014 7.6 0.79 0.56 0.35 0.01 0.024 0.2 0.065 0.068 ... 0.094

cv,%

low

high

2.9 17.5 2.7 14.7 >> 100 >> 100 1.3 17.9 17.4 6.4 4.8 8 3.8 1 4.2 55.1 5.6 ... 1

0.0307 5.79 80.2 9.86 170 1.7 13.71 0.0043 23.7 8.596 9.391 2.9 0.212

0.0434 43.1 107 57.2 >> 108 >> 106 15.43 0.17 995 16.9 14.76 6.29 0.314

note

(a) (a)

(a)

Only the lower limit of this parameter could be determined by the profile-t search method [9]. The upper limit approaches infinity.

Table S4: Best-fit model parameters determined from data depicted in Figure S2. actually be. This is the same as saying that the NAD+ substrate inhibition step can be described as instantaneous equilibration between E·P and E·P·N in Figure S1. In the third category are the two rate constants that characterize product inhibition by NADH (k4 and k−4 ). The likelihood profiles [4] for these two rate constants were closed from below not only at the 5% ∆SSQ level but also the 10% and 25% ∆SSQ levels (results not shown). However, the likelihood profiles showed no minimum at all and were perfectly flat to the right of the best-fit values. This can be seen in Figure S4. The interpretation of one-sided but otherwise perfectly flat likelihood profiles [4], such as those displayed in Figure S4, is straightforward. All one can say is that k4 must be greater than 170 s−1 (see Table S4) but there is no way of saying just high high k4 could actually be. Similarly, k−4 must be greater than 1.7 × 106 M−1 .s−1 (see Table S4) but the upper limit is undefined. This is the same as saying that the NADH product inhibition step can be described as instantaneous equilibration between E·P and E·P·NH in Figure S1. Rate constant correlations Figure S5 shows a plot of all pairs of rate constant values that lie within their respective confidence intervals as shown in Figure S4 (for pairs of k4 and k−4 ) and Figure S4 (for pairs of k7 and k−7 ). These correlation plots show that although the individual values of rate constants are poorly defined by the pre-steady state kinetic data, their ratios i.e. the corresponding dissociation equilibrium constants are very well defined. For example, in the left-hand panel of Figure S5 the plausible values of either k4 or k−4 span from the lower-left corner of the graph (i.e., from the well defined lower limit of the confidence 10

1.05

search bounds best-fit

1

1.2

SSQrel

1.4

search bounds best-fit

1

SSQrel

empirical interval, 5 % SSQ increase 1.1

empirical interval, 5 % SSQ increase

0.03

0.035

0.04

0.045

20

k2

40

60

k-2

Figure S3: Empirical likelihood profile for microscopic rate constants k2 and k−2 . For details see text. intervals) up to values at least four orders of magnitude higher (see also Figure S4). However, the plot of all plausible values of log(k4 ) against all plausible values of log(k−4 ) is a perfect straight line with a unit slope. Thus the ratio Kd(EP.NH) ≡ k4 /k−4 is invariant within the confidence intervals for either rate constant and it is equal to Kd(EP.NH) = 100 µM. Similarly all plausible values of the ratio Kd(EP.N) ≡ k−7 /k7 lie between 5.3 and 5.7 mM. Thus, in summary, in the analysis of one of the 16 combinatorial replicates depicted in Figure S2 we obtained well defined values of: • • • •

the lower limits for rate constant k4 and k−4 ; the lower limits for rate constant k7 and k−7 ; the equilibrium constant Kd(EP.NH) = k4 / k−4 = 100 µM; and the equilibrium constant Kd(EP.N) = k−7 / k7 = 5.5 mM.

2.3.2. Combinatorial replicates To our knowledge it is highly unusual that more than more than 10 microscopic rate constants should be uniquely determined from a single pre-steady state kinetic data set. In fact, we are not aware of any published report to that effect. Therefore, in order to stress-test our regression model in terms of reproducibility, we have performed 16 replicated regression analyses, by utilizing the combinatorial “mix-and-match” method described above. It should be noted that all series of 7×12 = 84 stopped-flow runs (7 varied NAD+ concentrations, 12 replicated injections to generate a single curve) were performed on different days, starting from fresh stock solutions. The bestfit values of all 13 adjustable rate constants are summarized in Figure S6. The numbering of combinatorial replicates on the horizontal axis corresponds to the rows of Table S2. The upper left panel in Figure S6 groups together all microscopic rate constants that appear in the substrate branch of the kinetic scheme in Figure S1, except for the NADH product inhibition rate constants k4 and k−4 and k7 and k−7 . The lower right panel groups together the best-fit values the four microscopic rate constants that characterize the interactions of A110. The results shown in Figure S6 indicate that the best-fit values of all microscopic rate constants except k4 and k−4 are very well reproduced across all 16 combinatorial replicates. The 11

1.1

empirical interval, 5 % SSQ increase

empirical interval, 5 % SSQ increase search bounds best-fit

1.04

SSQrel

1

1

1.02

1.05

SSQrel

1.06

1.08

search bounds best-fit

100

1000 10000 1000001e+0061e+0071e+008

1

10

100

1000 10000 1000001e+006

k4

k-4

empirical interval, 5 % SSQ increase

1.05

SSQrel

1.15

search bounds best-fit

1.1

1.15 1.1

1

1

1.05

SSQrel

empirical interval, 5 % SSQ increase search bounds best-fit

0.01

0.1

100

k7

1000

k-7

Figure S4: Empirical likelihood profile for microscopic rate constants k4 , k−4 (top) and k7 , k−7 (bottom). For details see text. coefficient of variation ranges from less than 5% for the forward rate constants (k2 , k3 and k5 ) to approximately 20% for the reverse rate constants (k−2 and k−3 ). Interestingly the best-fit values of k7 and k−7 are also very well reproduced. The coefficient of variation from replicates (n = 16) is approximately 15% for both k7 and k−7 , despite the fact that the upper limit of the confidence interval is undefined at the stringent 10% ∆SSQ level. In contrast, the replicated best-fit values of k4 and k−4 span at least two orders of magnitude. This observation is in agreement with the fact that neither k4 nor k−4 could be identified from individual replicates (see Table S4 and Figure S4. However, as before, the ratio k4 /k−4 shown in green in the lower left panel of Figure S6 is well defined by the data. The average (n = 16) value of Kd(EP.NH) ≡ k4 /k−4 is 100 µM. 2.3.3. Partially constrained model: Rapid equilibrium The results thus far indicate that all 16 combinatorial replicates produced only a small variation in the best-fit values of the NAD+ substrate inhibition rate constants k7 and k−7 . However, under stringent identifiability analysis [4] requirements based on Johnson’s 10% or 25% ∆SSQ 12

1000

k-7, s-1

100

1000 100

search k5 search k-5

1

10

k-4, µM-1 s-1

10000

search k3 search k-3

100

1000

10000

100000

1e+006

0.01

k4, s-1

0.1

k7, µM-1 s-1

Figure S5: Correlated values of rate constants that lie within the respective confidence intervals. Left: correlation between k4 and k−4 . Right: correlation between k7 and k−7 . For details see text. empirical rule [13], the upper limit is undefined for either rate constant. Faced with this apparent contradiction5 we have taken the more conservative view of the pre-steady state kinetic results and proceeded to investigate the mechanistic scenario whereby both substrate inhibition by NAD+ and product inhibition by NADH are under rapid equilibrium. The main question is whether or not by setting the association rate constants k−4 and k7 to arbitrary “high” values might influence the best-fit values of the remaining microscopic rate constants. To answer this question, a series of five separate fitting models was constructed (five variants for for each of the 16 combinatorial replicates) where the bimolecular association rate constants k−4 and k7 were both set to an identical “high” values and subsequently held fixed in the regression analysis. To arrange for type constrained regression analysis, the [constants] section in the script listed in Appendix A.1 was modified by removing the question marks next to k−4 and k7 . For example, to set kon = k−4 = k7 = 10 µM−1 .s−1 , we used the coding below: [constants] k2 = 0.03 ? k3 = 80 ? k4 = 1000 ? k5 = 10 ? k7 = 10 k8 = 10 ? k9 = 5 ?

, , ,

k-2 = 10 ? k-3 = 40 ? k-4 = 10

, , ,

k-7 = 50000 ? k-8 = 10 ? k-9 = 0.3 ?

; fixed "k-4" ; fixed "k7"

The particular fixed values were kon = k−4 = k7 = 10, 20, 50, 100, 1000 µM−1 .s−1 , i.e., spanning from 107 to 109 M−1 .s−1 . In either of the five cases, the best-fit model curves overlaid on the experimental data as well as the corresponding residual plots were visually indistinguishable from Figure S2 (results not shown). As expected from the confidence interval profile shown in 5 It is possible that the apparent very high reproducibility of the best-fit values of both k and k 7 −7 is some type of artifact given that the upper limits do not exist under stringent identifiability requirements.

13

+

NAD product inhibition

0

2

k7 k-7 k7/k-7

log10 (rate constant)

0

1

k2 k-2 k3 k-3 k5

-2

-1

log10 (rate constant)

2

4

catalytic cycle except NADH product inhibition

5

10

15

5

replicate no. NADH product inhibition

15

A110 inhibition

0

0.5

1

k8 k-8 k9 k-9

-0.5

4

log10 (rate constant)

6

k4 k-4 k4/k-4

2

log10 (rate constant)

10

replicate no.

5

10

15

5

replicate no.

10

15

replicate no.

Figure S6: Reproducibility of rate constant determination in 16 combinatorial replicates. For details see text. Figure S4, the best-fit residual sum of squares was almost exactly 5% higher compare to the first round of analysis where both k7 and k−7 were treated as adjustable model parameters. The numerical results are summarized in Table S5. The results for kon = 1000 µM−1 .s−1 were exactly identical to those obtained for kon = 100 µM−1 .s−1 and are not shown. The results summarized in Table S5 indicate that the average (n = 16) best-fit values of microscopic rate constants appearing Figure S1 are largely insensitive to the assumed, fixed values of the association rate constants k−4 and k7 . The relative changes going from kon = 10 µM−1 .s−1 to kon = 100 µM−1 .s−1 amount to at most a few percentage points. The same applies to the various rate constant ratios (i.e., equilibrium constants) listed at the bottom of Table S5. In conclusion, the minimal partially constrained kinetic model for the pre-steady state kinetic data is one where both the product inhibition by NADH and substrate inhibition by NAD+ should be considered to occur under rapid equilibrium approximation. Any particular value of kon = 14

unit

kon = 10 µM−1 .s−1

kon = 20 µM−1 .s−1

kon = 50 µM−1 .s−1

kon = 100 µM−1 .s−1

k2 k−2 k3 k−3 k4 k−4 k5 k7 k−7 k8 k−8 k9 k−9

µM−1 .s−1 s−1 s−1 s−1 s−1 µM−1 .s−1 s−1 µM−1 .s−1 s−1 µM−1 .s−1 s−1 µM−1 .s−1 s−1

0.0315 10.7 84.5 47.4 1150

0.0317 10.8 83.1 45.5 2300

0.0318 10.9 82.3 44.6 5730

0.0318 10.9 82.1 44.3 11500

k−2 /k2 k3 /k−3 k4 /k−4 k−7 /k7 k−8 /k8 k−9 /k9

µM – µM µM µM µM

13.7 56000 13.9 10.9 5.65 0.271

± 0.0014 ± 2.9 ± 2.1 ± 9.7 ± 160 10 ± 0.26 10 ± 2700 ± 0.94 ± 0.4 ± 0.74 ± 0.004

338 1.78 115 5600 0.788 0.0479

± ± ± ± ± 20 13.7 ± 20 113000 ± 13.9 ± 11.0 ± 5.58 ± 0.270 ±

0.0014 2.9 1.9 9.5 330 0.25 5400 0.94 0.4 0.73 0.004

342 1.82 115 5630 0.792 0.0485

± ± ± ± ± 50 13.6 ± 50 283000 ± 13.9 ± 11.0 ± 5.53 ± 0.270 ±

0.0014 2.9 1.9 9.3 810 0.25 14000 0.94 0.4 0.72 0.004

344 1.85 115 5650 0.795 0.0488

± 0.0014 ± 2.9 ± 1.9 ± 9.3 ± 1600 100 13.6 ± 0.25 100 566000 ± 27000 13.9 ± 0.94 11.0 ± 0.4 5.51 ± 0.72 0.270 ± 0.004 344 1.85 115 5660 0.796 0.0489

Table S5: Averages and standard deviations from combinatorial replicates (n = 16) for microscopic rate constants appearing in Figure S1 while holding the bimolecular association rate constants k−4 and k7 at various fixed values. k−4 = k7 higher than approximately 107 M−1 .s−1 provides an equally good description of the experimental data. The best-fit values of the remaining microscopic rate rate constants appearing in Figure S1 are insensitive to the arbitrary choice of kon > 107 M−1 .s−1 . This rapid-equilibrium constrained kinetic model could be graphically represented as is shown in Figure S7. The dashed arrows represent equilibrium binding steps, for which only the lower limit of kon can be estimated. 2.4. Pre-steady state kinetics: Summary 1. The inhibitor A110 binds predominantly to an enzyme–product complex, E˜ P. 2. E˜ P is either the covalent intermediate or the noncovalent E·XMP complex. 3. The association rate constant for I + E˜ P → E˜ P·I is 5.5 × 106 M−1 .s−1 . 4. The dissociation rate constant for E˜ P·I → I + E˜ P is 0.27 s−1 (t1/2 = 2.5 sec). 5. The corresponding Kd is 49 nM. 6. The inhibitor A110 binds more weakly to the enzyme–substrate complex, E·A. 7. The association rate constant for I + E·A → E·A·I is 1.4 × 107 M−1 .s−1 . 8. The dissociation rate constant for E·A·I → I + E·A is 11 s−1 (t1/2 = 0.063 sec). 9. The corresponding Kd is 0.80 µM. 10. B dissociates rapidly from E·A·B (k−2 = 11 s−1 ; not a “sticky” substrate). 11. The forward hydride transfer rate constants is k3 = 80 s−1 (k3 > k−2 ). This necessitates the steady state approximation in the analysis of initial rates. 12. The reverse hydride transfer rate constants is k−3 = 40 s−1 (Keq = k3 /k−3 ≈ 2). The forward hydride transfer is slightly favored over the reverse transfer. 15

kon > 106 M-1s-1

E.B

E~P.B Kd(EP.B) +B

Kd(EP.Q)

k3 E

E.A.B k2

E~P

E-P.Q k-3

+Q +I

+B

k-2 E.A

k-9

k5

E~P.I

+I

k-8

k8

+A saturating

k9

P

E.A.I

Figure S7: Minimal partially rapid-equilibrium kinetic mechanism for the the inhibition BaIMPDH by A110. The steps shown in red are assumed to be under rapid rapid equilibrium. 13. Hydrolysis of the covalent intermediate and product release is partially rate limiting (k5 = 14 s−1 , k5 < k3 for hydride transfer). 14. NADH (Q) is a product inhibitor, with predicted Kd ≈ 100 µM (rapid equilibrium). 15. NAD+ (B) is a substrate inhibitor, with predicted Kd ≈ 6 mM (rapid equilibrium). 3. Steady-state initial rate kinetics We utilized steady-state initial rate kinetic experiments mainly to check the validity of our primary results, which are based on the stopped-flow rapid kinetic experiments (see section 2 above). 3.1. Experimental data The steady-state initial rate data were organized in the same fashion, and obtained under the same reaction conditions, as the stopped-flow transient kinetics data discussed above. Four categories of experiments were perfomed at a saturating concentration of the substrate IMP ([S]0 = 1.0 mM): 1. 2. 3. 4.

NAD+ varied alone, in the absence of any additional component. NAD+ varied in the presence of various amounts of NADH (product inhibition). NAD+ varied in the presence of various amounts of A110. NAD+ varied in the presence of either 1 H- or 2 H-IMP.

The nominal concentration of the enzyme, [E]0 = 23 nM, was identical in all initial rate experiments. However, the amplitude (Vmax ) of presumably identical experiments repeated after an extended period of time showed some non-negligible variations, approximately 10%–20%. Therefore the actual enzyme concentration in the global fit of initial rate data was treated as an adjustable model parameter. 16

Finally, one additional initial rate data set (the “double-inhibition” experiment) was collected at sub-saturating concentration of the IMP substrate ([S]0 = 0.2 mM) and therefore it is not fully compatible with the transient kinetic experiments. In that special case, both A110 and XMP were both varied simultaneously as inhibitors. 3.2. Theoretical model 3.2.1. “Classical” rate equation (King-Altman & Cleland) In order to facilitate the derivation of the “classical” initial rate equation (i.e., one in which possible inhibitor depletion due to “tight binding” conditions is ignored), the DynaFit software [2] was presented with mechanism specification listed below: [task] ... data = rates approximation = king-altman [mechanism] enzyme E.A reaction B ---> P + Q modifiers I E.A + B E.A.B : k2 E.A.B E.P.Q : k3 E.P.Q E.P + Q : k4 E.P ---> E.A + P : k5 E.P + B E.P.B : k7 E.A + I E.A.I : k8 E.P + I E.P.I : k9

k-2 k-3 k-4 k-7 k-8 k-9

This input text corresponds to the kinetic mechanism shown in Figure S1. Note that under IMP saturating conditions the “free enzyme” species is effectively the enzyme–substrate complex E.A. Also note that under IMP saturation the kinetic mechanism turn to Uni Bi, with effectively only a single substrate (B). These facts are indicated by the following input lines: ... enzyme E.A reaction B ---> P + Q ... When DynaFit processed the input text listed above (see Appendix A.2.1), it automatically generated the following algebraic expressions and utilized those to perform the least-squares fit of experimentally observed initial rate data: v = [E]0 N

N D

(S13)

= n1 [B]

(S14)

D = d1 + d2 [I] + d3 [Q] + d4 [B] + d5 [Q][I] + d6 [B][I] + d7 [B][Q] +d8 [B]2

(S15) 17

k2 k3 k4 k−2 k−3 + k−2 k4 + k3 k4

n1

=

d1

=

1

(S17)

d2

=

k8 k−8

(S18)

d3

=

k−2 k−3 k−4 k5 (k−2 k−3 + k−2 k4 + k3 k4 )

(S19)

d4

=

k2 (k−3 k5 + k4 k5 + k3 k5 + k3 k4 ) k5 (k−2 k−3 + k−2 k4 + k3 k4 )

(S20)

d5

=

k−2 k−3 k−4 k8 k5 k−8 (k−2 k−3 + k−2 k4 + k3 k4 )

(S21)

d6

=

k2 k3 k4 k9 (k k5 k−9 −2 k−3 + k−2 k4 + k3 k4 )

(S22)

d7

=

k2 k−4 (k−3 + k3 ) k5 (k−2 k−3 + k−2 k4 + k3 k4 )

(S23)

d8

=

k2 k3 k4 k7 k5 k−7 (k−2 k−3 + k−2 k4 + k3 k4 )

(S24)

kcat =

n1 d4

=

k3 k4 k5 k−3 k5 + k4 k5 + k3 k5 + k3 k4

(S25)

Km(B) =

d1 d4

=

k5 (k−2 k−3 + k−2 k4 + k3 k4 ) k2 (k−3 k5 + k4 k5 + k3 k5 + k3 k4 )

(S26)

kcat n1 = Km(B) d1

=

k2 k3 k4 k−2 k−3 + k−2 k4 + k3 k4

(S27)

18

(S16)

Ki(I) =

d1 d2

=

k−8 k8

(S28)

Ki(I,B) =

d4 d6

=

k−9 (k−3 k5 + k4 k5 + k3 k5 + k3 k4 ) k3 k4 k9

(S29)

Ki(B,Q) =

d3 d7

=

k−2 k−3 k2 (k−3 + k3 )

(S30)

Ki(B) =

d4 d8

=

k−7 (k−3 k5 + k4 k5 + k3 k5 + k3 k4 ) k3 k4 k7

(S31)

Ki(Q) =

d1 d3

=

k5 (k−2 k−3 + k−2 k4 + k3 k4 ) k−2 k−3 k−4

(S32)

Ki(Q,B) =

d4 d7

=

k−3 k5 + k4 k5 + k3 k5 + k3 k4 k−4 (k−3 + k3 )

(S33)

It is noteworthy that only one of the two A110 inhibition constants (the “competitive” constant Ki(I) ) is by definition equivalent to the equilibrium dissociation constant of the corresponding enzyme–inhibitor complex:

Kd(EA.I)

≡

k−8 k8

Ki(I)

=

k−8 k8

In contrast the “uncompetitive” inhibition constant Ki(I,B) is by definition larger than the corresponding equilibrium constant, as shown in Eqn (S34). See Figure S8 for graphical illustration of which microscopic rate constant enter into the definition of Ki(I,B) .

Kd(EP.I)

≡

k−9 k9

Ki(I,B)

=

k−9 k3 + k−3 + k4 1 + k5 k9 k3 k4

! (S34)

This result is contrary to assumptions that occasionally appear in the biochemical literature. For example, Biswanger [p. 3][15] writes in a well-regarded enzyme kinetics textbook: “From equilibrium treatments thermodynamic constants, like association or dissociation constants, are derived, while kinetic studies yield the more complex kinetic constants. On the other hand there are also similarities. [...] [F]or example inhibition constants, although determined kinetically, are really dissociation constants.” 19

E.A.I

E.A.I =

Kd E.A

Ki(I)

Ki(I,B) E-P.Q

E.A.B

E.A

E-P

E.A.B

E-P.B

E-P.Q

E-P

E-P.B

Kd E-P.I

E-P.I

Figure S8: The red arrows represent all microscopic rate constant that define composite inhibition constant for the kinetic mechanism displayed in Figure S1. The “competitive” Ki is equal to the corresponding Kd , whereas the “uncompetitive” Ki is not. The IMPDH inhibition mechanism shown in Figure S1 provides a counter-example illustrating clearly that not all inhibition constants “are really dissociation constants”, although some are. The computer-generated initial rate model shown in Eqns (S13)–(S24), formulated in terms of the microscopic rate constants, was converted by standard algebraic manipulations [16, pp. 523-530] into the more conventional form in terms of the composite “kinetic constants”. The resulting steady-state initial rate law is shown in Eqn (S35), where n and d represent the numerator and denominator defined by Eqn (S36) and Eqn (S37), respectively. v =

[E]0 kcat

n

=

[B] Km(B)

d

=

1+

+

n d

(S35) (S36)

[I] [Q] [B] [Q][I] + + + Ki(I) Ki(Q) Km(B) Ki(Q) Ki(I)

[B][I] [B][Q] [B]2 + + Km(B) Ki(I,B) Ki(B,Q) Ki(Q) Km(B) Ki(B)

(S37)

The kinetic constants kcat , Km(N) , etc., are defined in terms of the microscopic rate constants appearing in Figure S1 as shown in Eqns (S25)–(S33). However, note that there is an inherent ambiguity in the definition of inhibition constants, as explained by Segel [p. 524][16]: “[Inhibition] constants associated with a given ligand will equal the ratio of two denominator coefficients. The ratio must be chosen so that, after canceling subscript letters, the letter corresponding to the ligand associated with the constant remains as a subscript in the denominator of the ratio. [...] For some systems, it will be possible to find quite a few ratios of coefficients that satisfy the cancelation requirements above.” 20

Indeed, in the case of the kinetic mechanism depicted in Figure S1 we have to choose between two mutually exclusive formulations of the initial rate equation. One variant is shown immediately above. Another variant is defined by Eqn (S38), where d0 is an alternate definition of the denominator.

d0

=

1+ +

=

[B] [B][Q] [B]2 [Q] + + + Ki(Q) Km(B) Km(B) Ki(Q,B) Km(B) Ki(B)

[I] [Q][I] [B][I] + + Ki(I) Ki(Q) Ki(I) Km(B) Ki(I,B)

! [B] [B] [Q] [Q] 1+ + 1+ + Ki(Q) Km(B) Ki(Q,B) Ki(B) " ! # 1 [Q] 1 [B] +[I] 1+ + Ki(I) Ki(Q) Ki(I,B) Km(B)

(S38)

The difference between the two formulations of the IMPDH rate equation appears in the particular denominator term that contains the concentration product [B][Q]. In the case of denominator d defined by Eqn (S37), NAD+ (B) is formally treated as a “mixed type” inhibitor because it is associated with two inhibition constants, Ki(B) and Ki(B,Q) , whereas NADH (Q) is treated as an “uncompetitive” inhibitor because it is associated with only a single inhibition constant Ki(Q) . In contrast, in the case of denominator d0 defined by Eqn (S38), NAD+ is treated as an “uncompetitive” inhibitor because it is associated with a single inhibition constants, Ki(B) , whereas NADH is treated as a “mixed-type” inhibitor, because it is associated with two inhibition constant Ki(Q) and Ki(Q,B) . These two view of the same kinetic process are both equally correct, but they are mutually exclusive. Therefore it is meaningless to ask whether NAD+ substrate inhibition in IMPDH kinetics should be classified as uncompetitive or mixed-type, or equivalently whether NADH product inhibition is uncompetitive or mixed-type. The only appropriate answer is, “it depends” which of the denominator formulations (d or d0 ) is chosen and, importantly, that choice is entirely arbitrary. 3.2.2. “Tight binding” rate equation (Morrison) All steady-state initial rate experiments were conducted at nominal enzyme concentrations 23 nM, based on Bradford assay. The equilibrium dissociation constant k−9 /k9 predicted from the pre-steady state kinetic results is approximately 50 nM (Table S5). When the active enzyme concentration and the inhibition constants are comparable in magnitude, “tight binding” (i.e. inhibitor depletion) must be properly taken into account in the mathematical model for the analysis of initial rates. Morrison [17] derived a generic form of the enzymatic initial rate law that applies to all enzyme mechanisms where the inhibitor forms any number of 1:1 catalytically inactive complexes with various enzyme forms. Morrison’s original formula6 is reproduced without changes in Eqn (S39), where [E]0 is the total or analytic concentration of enzyme active sites; [I]0 is the total 6

See Eqn. (12) in [17, p. 272].

21

or analytic concentration of the inhibitor; D is the denominator of the initial rate equation derived by the King-Altman method [18] in the absence of the inhibitor; N is the numerator of the initial rate equation derived by the King-Altman method in the absence of the inhibitor and assuming that that there is no product formation from any enzyme–substrate–inhibitor complex (i.e., disallowing partial inhibition); Ni is the numerator of the distribution equation for the given enzyme–inhibitor complex [19]; and Ki is the inhibition constant corresponding to the enzyme– inhibitor complex specified by the term Ni .

v =

v u  2   u  u u     u  u u u     u N  u [I]0 − [E]0  4 [E]0 1 [I] − [E] t 1   0 0  + !+ ! −  !+    (S39)    2   P Ni D D P Ni   P Ni  D    Ki Ki Ki

When the generic terms appearing in Eqn (S39) (i.e., N, D, Ni , and Ki ) are appropriately specialized on the basis of the assumed kinetic mechanism shown in Figure S1, we obtain Eqn (S40), where v0 is the uninhibited reaction rate (observed at zero inhibitor concentration) and Ki∗ is the apparent inhibition constant [20, 21]. The composite kinetic constants that appear in Eqns (S41)–(S42) are defined in terms of the microscopic rate constants appearing in Figure S1 as is shown in Eqns (S25)–(S33).

v = v0

v0

Ki∗

[E]0 − [I]0 −

= [E]0 kcat

=

Ki∗

q 2 + [E]0 − [I]0 − Ki∗ + 4 [E]0 Ki∗ 2 [E]0

[B] Km(B) [Q] [B] [Q] [B] 1+ + 1+ + Ki(Q) Km(B) Ki(Q,B) Ki(B)

[Q] [B] [Q] [B] 1+ + 1+ + Ki(Q) Km(B) Ki(Q,B) Ki(B) ! [Q] 1 [B] 1 1+ + Ki(I) Ki(Q) Ki(I,B) Km(B)

!

(S40)

(S41)

! (S42)

3.3. Steady-state initial rate results 3.3.1. Predicted vs. observed initial rates The fitting model for transient kinetic data contains 11 adjustable microscopic rate constants (Table S5) in addition to the fixed values of k−4 and k7 . The question arises whether or not the kinetic model with such large degree of flexibility can accurately predict steady state initial rates determined in a series of independent experiments under identical conditions. The DynaFit script listed in Appendix A.2.2 was designed to answer that question. 22

This script performs a highly constrained least-squares fit, such that only the enzyme concentrations are optimized whereas all composite kinetic constants are held fixed at values predicted from the results of stopped-flow experiments. In particular, the best-fit values of microscopic rate constants listed in the rightmost column of Table S5 were used to compute the predicted values of steady state kinetic constants by using Eqns (S25)–(S33). These predicted values are listed in Table 2, main manuscript, column labeled “predicted”. The regression model proper is the “Morrison” Eqn (S40). Since all kinetic constants are fixed in the highly constrained regression analysis, the “fit” largely a comparison between initial rates predicted from the stopped-flow experiment and initial rates observed experimentally. The results are shown graphically in Figure 3 of the main manuscript. The best-fit values or all adjustable enzyme concentrations are listed in Table S6. #

par/set

1 2 3 4 5

[E](1) 0 , µM [E](2) 0 , µM [E](3) 0 , µM [E](4) 0 , µM [E](5) 0 , µM

inital 0.01 0.01 0.01 0.01 0.01

final ± std.err. 0.00978 0.00838 0.00921 0.01067 0.00910

± ± ± ± ±

0.00007 0.00012 0.00013 0.00010 0.00010

cv,%

Note

0.8 1.5 1.5 1.0 1.1

Table S6: Global fit of combined initial rate data to the system of Eqns (S40)–(S42). All steadystate kinetic constants were held fixed at values predicted by the analysis of stopped-flow data. For further details see text. The results displayed graphically in Figure 3 (main manuscript) show that the 13 microscopic rate constants determined by the stopped-flow measurements predict the initial rate data reasonably well. For example the position of the maximum on the NAD+ saturation curve (approximately at [NAD+ ] = 1.5 mM), in the upper left panel, is very well predicted, as is the slope of the downward portion due to substrate inhibition. In the upper right panel of Figure 3 (main manuscript), the relative spacing of the NAD+ saturation curves at different concentration of NADH as product inhibitor is also very well predicted. The lower left panel shows the inhibition effect of A110 at relatively low concentrations of NAD+ . Again the relative spacing of the substrate saturation curves is very well predicted. The same applies to the right-hand panel of Figure 3 (main manuscript) displaying the inhibitory effect of A110 at relatively high concentrations of NAD+ . 3.3.2. Predicted vs. observed kinetic constants In the next round of kinetic analysis, the experimental data displayed in Figure 3 (main manuscript) were subjected to a least-squares fit in which the kinetic constants defined in terms of the microscopic rate constants by Eqns (S25)–(S33) were treated as optimized parameters. The required DynaFit notation is exactly identical to the listing in Appendix A.2.2, except for the fact that the initial estimates of model parameters are marked with question marks: ; ; Enzyme concentrations in different experiments: First one fixed. ; cE1 = 0.009782 ; based on ’compare-v’ script cE2 = 0.01 ?

23

cE3 = 0.01 ? cE4 = 0.01 ? cE5 = 0.01 ? ; ; Extinction coefficient of NADH: ; rQ = 6.22 ; fixed ! ; ; Predicted kinetic constants: Get confidence intervals. ; kcat = 11.6 ?? KmB = 415 ?? KiI = 0.791 ?? KiIB = 0.0572 ?? KiB = 6610 ?? KiQ = 301 ?? KiQB = 87.2 ??

A single question mark identifies a model parameter that should be optimized in the regression analysis. A double question mark identifies a parameter, for which we additionally desire the asymmetric confidence interval to be computed by using the profile-t method of Bates & Watts [9–11] Note that one of the enzyme concentration ([E](1) 0 = 9.78 nM) was treated as a fixed parameter set to the best fit value determined in the previous round of analysis, Table S6. This was necessary to “anchor” the kcat value because the product Vmax = kcat × [E]0 appears in the initial rate equation and therefore it is impossible to simultaneously determine both kcat and [E]0 from any particular set of initial rate measurements. The results of fit are shown graphically in Figure S9. The list of best-fit adjustable model parameters is shown in Table 2, main manuscript. The column labeled “predicted” lists the expected values of kinetic constants, i.e., the values computed beforehand using Eqns (S40)–(S42) and the numerical values of stopped-flow microscopic rate constants listed in the right-most column of Table S5. The two sets of kinetic constant values (predicted from transient kinetics vs. observed from initial rates) are in good agreement within the experimental error expressed as the 90% confidence-level intervals. 3.3.3. Diagnostic Eadie-Hofstee plots The Eadie-Hofstee plots corresponding to Figure S9 are shown in Figure S10. To arrange for automatic generation of Eadie-Hofstee plots by the DynaFit software, the input script listed in Appendix A.2.2 was modified simply by inserting a single line into the [data] section, as follows: [data] transform EH

; P + NH E.S + N ---> E.S.N E.S.N E.P.NH E.P.NH E.P + NH E.P ---> E.S + P E.P + N E.P.N

: : : : :

k1 k2 k3 k4 k5

26

k-2 k-3 k-5

v = [E]0

#

par/set

initial

1 2 3 4 5 6

Km / 1 Ki / 1 kK / 1 Km / 2 Ki / 2 kK / 2

500 5000 0.01 500 5000 0.01

n1 [N] d1 + d2 [N] + d3 [N][NH] + d4 [N]2

(S43)

d1

= 1

(S44)

d2

=

k1 (k−2 k4 + k3 k4 + k2 k4 + k2 k3 ) k2 k3 k4

(S45)

d3

=

k1 k−3 (k−2 + k2 ) k2 k3 k4

(S46)

d4

=

k1 k5 k4 k−5

(S47)

kcat(f) =

n1 d2

=

k2 k3 k4 k−2 k4 + k3 k4 + k2 k4 + k2 k3

(S48)

Km =

d1 d2

=

k2 k3 k4 k1 (k−2 k4 + k3 k4 + k2 k4 + k2 k3 )

(S49)

kcat n1 = Km d1

=

k1

(S50)

Ki(N) =

d2 d4

=

k−5 (k−2 k4 + k3 k4 + k2 k4 + k2 k3 ) k2 k3 k5

(S51)

Ki(NH,N) =

d2 d3

=

k−2 k4 + k3 k4 + k2 k4 + k2 k3 k−3 (k−2 + k2 )

(S52)

final ± std.err. 500 7000 0.0247 1030 7800 0.0104

± ± ± ± ± ±

30 600 0.0008 90 900 0.0004

cv,%

low

high

6.0 7.9 3.4 8.4 11.0 4.2

440 5900 0.0229 870 6200 0.0095

560 8300 0.0265 1240 9800 0.0114

note

Table S7: Results of fit from the analysis of kinetic isotope effect. All experiments were performed in the same session (i.e., identical enzyme concentration) at saturating [IMP] = 1 mM. The notation "/1" refers to results obtained with 1 H-IMP, whereas "/2" refers to results obtained with 2 H-IMP. The optimized parameter kK is defined as the corresponding kcat /Km . Asymmetric confidence interval bounds “low” and “high” are at the 95% confidence level. Note that in this simplified kinetic mechanism the specificity constant kcat /Km is by definition equal to the microscopic association rate constant k1 . More generally, kcat /Km values always include only those microscopic rate constants up to (and including) the first irreversible step 27

1 H-IMP 2

0.2 0

v, mOD/sec

0.4

H-IMP

0

2000

4000

[NAD+], µM

6000

(app)

8000

(app)

Figure S11: Determination of Km and kcat (assuming [E]0 = 10 nM) for NAD+ at saturating concentration (1.0 mM) of either 1 H-IMP or 2 H-IMP, as indicated in the figure margin. [22]. Therefore, if NAD+ binding were in fact “sticky”, then the experimentally observed kcat /Km value should be entirely insensitive to isotopic substitution in 1 H-IMP vs. 2 H-IMP. To test whether k−1 is effectively zero or nonzero (as is suggested by the stopped-flow experiments) we measured kcat /Km either with 1 H-IMP or with 2 H-IMP at saturating concentrations. The regression model was Eqn (S53), where vobs is the observed initial rate in mOD/sec; rQ is the molar response coefficient of NADH (6.22 mOD/µM); [E]0 is the active enzyme concentration (10 nM); Km is the Michaelis constant; Ki is the substrate inhibition constant; and kK is kcat /Km for the given isotope. Here we utilized Northrop’s [23] rearrangement of the Michaelis-Menten equation (adjusted for the substrate inhibition term containing Ki ). The reason for recasting the rate equation in this particular way is that, as was documented in ref. [23], the algebraic rearrangement significantly increases the accuracy of specifically kcat /Km determinations. The requisite DynaFit script is listed in Appendix A.2.3. vobs

[S]0 Km + [S]0 + [S]20 /Ki ! Km [S]0 kcat Km Km + [S]0 + [S]20 /Ki

= rQ [E]0 kcat

= rQ [E]0

= rQ [E]0 kK kK

≡

kcat Km

Km [S]0 Km + [S]0 (1 + [S]0 /Ki )

(S53) (S54)

28

The results of fit are summarized graphically in Figure S11. The best-fit model parameter values are listed in Table S7. Notation “/1" and “/2" refers to 1 H-IMP vs. 2 H-IMP, respectively. Columns “low” and “high” contain the 95% confidence intervals computed by the profilet method of Bates & Watts [9, 11]. The best-fit parameter values listed in Table S7 show that for 1 H-IMP kK ≡ kcat /Km = (0.0247 ± 0.0008) µM−1 .s−1 whereas for 2 H-IMP kK ≡ kcat /Km = (0.0104 ± 0.0004) µM−1 .s−1 . Thus the observed kinetic isotope effect is both surprisingly large and also highly statistically significant. Note that the coefficient of variation for both 1 H-IMP and 2 H-IMP determination of kK is lower than 5%. This high level of precision would not be possible to achieve with the classical as opposed to the rearranged [23] algebraic form of the MichaelisMenten equation. Using the standard formula for statistical uncertainty of a ratio from error propagation theory [24, p. 62, Eqn 4-11], the most probable value and the standard error7 of the kcat /Km isotope effect can be computed as

D

kcat Km

! =

s  !2 !2  0.0247  0.0008 0.0004  + 1 ±  0.0104  0.0247 0.0104 

= 2.38 ± 0.12 Assuming that the intrinsic isotope effect on hydrogen transfer in either direction is equal to approximately D k3 =D k−3 = 4.0, the predicted value of the observed isotope effect on kcat /Km is approximately D (kcat /Km ) = 1.34. This value was calculated using the microscopic rate constants listed in Table S5 applied to the system of Eqns (S25)–(S33). Thus the observed isotope effect is even significantly larger than predicted, which provides further support to the conclusion that NAD+ is not a “sticky” substrate for BaIMPDH∆L in contrast with other IMPDH isoforms [5]. 3.3.5. Double inhibition experiment Experimental design and preliminary data The results of both the stopped-flow kinetic investigations as well as steady-state initial rate measurements described thus far clearly indicate that the inhibitor A110 binds strongly (Kd ≈ 50 nM) either to the covalent intermediate E-P, or to the noncovalent complex E.P, or to both of these enzyme species. Significantly weaker binding (Kd ≈ 0.7 µM) is observed to the enzyme– substrate complex E.A. Under IMP saturating conditions it is impossible to discriminate between these alternate scenario or asses their relative contributions. The experiments described and analyzed in this section were designed to decide whether or not A110 binds preferentially either to E-P or to E.P. The experimental design relies on a doubleinhibitor assay involving A110 as the inhibitor of interest and XMP (P) as product inhibitor. The experimental design for this semi-quantitative study was as follows: • • • •

The enzyme concentration was [E] = 10 nM. The IMP substrate concentration was [A] = 0.2 mM. The NAD+ cofactor concentration was [B] = 1.5 mM. The A110 inhibitor was varied as [I] = 0, 15, 30, 60, 120, 180 nM

7 Note that the covariance term in Bevington’s Eqn 4-11 can be neglected because the two determinations of k /K cat m are statistically independent, having originated in separate experiments.

29

• XMP as product inhibitor was varied as [P] = 0, 0.125, 0.25, 0.5, 1, 2 mM • At each given [XMP] concentration, the apparent inhibition constant for A110, Ki∗ , was determined by a fit of [I] vs. initial rate data to the Morrison Eqn (S40). • A plot of Ki∗ vs. [XMP] was constructed to ascertain whether Ki∗ changes with [XMP] and, if it does, in what fashion. The basic premise of the double-inhibition experiment is that if Ki∗ for A110 were to show a detectable decrease (stronger binding) with increasing [XMP], it would be reasonable to conclude that A110 binds preferentially to the noncovalent complex E.P. The reasoning is that at elevated levels of [XMP], more E.P is available for A110 binding and therefore the inhibition potency would appear enhanced. In contrast, if Ki∗ for A110 were to show a detectable increase (weaker binding) with increasing [XMP], it would be reasonable to conclude that A110 binds preferentially to the covalent intermediate E-P. In order to allow for XMP product inhibition to be observable experimentally, the IMP substrate concentration was decreased from the saturating concentration [A] = 1.0 mM, used every(app) where else in this report, to [A] = 200 µM. The apparent Michaelis constant Km(A) at [B] = 1.5 mM is approximately 60 µM (see Appendix A.2.4 and Figure S12A). Therefore the [A] = 200 µM is clearly sub-saturating. A back-of-the-envelope calculation shows that under these conditions approximately 23% of the enzyme is the free, unliganded state. This allows XMP to express its product-inhibition potential. The apparent inhibition constant of XMP (P) at [A] = 200 µM and [B] = 1.5 mM is approximately 1.1 mM (see Appendix A.2.5 and Figure S12B). This allows XMP to affect easily detectable product inhibition within the concentration range chosen for the double-inhibitor experiment, [P]max = 2.0 mM. A

B XMP : Ki(app) = (1160 +/- 50) µM

0

0

0.2

0.4

v, mOD/sec

0.2

v, mOD/sec

0.6

0.4

0.8

IMP : Km(app) = (61 +/- 3) µM

0

500

1000

0

[IMP], µM

1000

[XMP], µM

(app)

(app)

Figure S12: A: Determination of Km(A) at [B] = 1.5 mM. B: Determination of Ki(P) at [A] = 200 µM and [B] = 1.5 mM.

30

2000

Experimental determination of Ki∗ vs. [P] Inhibitor dose-response curves observed at various fixed concentrations of [P] and [A] = 200 µM, [B] = 1.5 mM were fit to the Morrison Eqn (S40) to determine the apparent inhibition constant, Ki∗ . The requisite DynaFit script is listed in Appendix A.2.6. The results of fit are illustrated graphically in Figure 4A of the main manuscript. The best-fit values of the apparent inhibition constants vs. product concentrations are listed in Table S8. [XMP], µM 0 125 250 500 1000 2000

Ki∗ , µM ± Std. Error 0.141 0.153 0.145 0.163 0.224 0.333

± ± ± ± ± ±

0.007 0.008 0.008 0.011 0.023 0.058

Table S8: Experimentally observed dependence of the apparent inhibition constant, Ki∗ , for A110 at [IMP] = 200 µM and [NAD+ ] = 1.5 mM. Note in Table S8 that the formal standard error of determination increases with increased [XMP] concentration. This is a consequence of the fact that as the two highest values of [XMP] the apparent inhibition constant (Ki∗ > 200 nM) already increased above the highest concentration of A110 used in this experiment ([A110]max = 180 nM). If all inhibitor concentrations utilized in a dose-response experiment are lower than the apparent inhibition constant (often identical to the IC50 unless inhibitor depletion is involved), the precision and accuracy of Ki∗ is adversely affected. Theoretical prediction for the dependence of Ki∗ on [P] With the preliminary data in hand regarding inhibitory potency of XMP and the apparent Michaelis constant for IMP, it is possible set up a theoretical model for the double-inhibitor experiment, by invoking the kinetic mechanism shown in Figure S13. Note that, for simplicity in this semi-quantitative analysis, it was assumed that (i) the kinetic mechanism is effectively Bi Bi Ordered and (ii) that the inhibitor does not bind to the free unliganded enzyme. The microscopic rate constants k2 , k−2 , k3 , k−3 , k4 , k−4 , k5 , k7 , k−7 , k8 , k−8 , k9 , and k−9 were set to the values determined in the stopped-flow kinetic experiment (Table S5). Approximate values (app) of the microscopic rate constants k1 , k−1 were estimated on the basis of the Km(A) value shown in Figure S12 (left panel). Similarly, approximate values of the microscopic rate constants k6 , k−6 (app) were estimated on the basis of the Ki(P) value shown in Figure S12 (right panel). The focus in the heuristic simulation Ki∗ vs. [P] is mainly on the plausible values of the equilibrium constant k−10 /k10 , measures the binding affinity (if any) of A110 toward the noncovalent complex E.P. A DynaFit script containing the following input was utilized to derive steady-state initial rate equation corresponding to the kinetic mechanism in Figure S13: [task] ... data = rates approximation = king-altman 31

E-P.B E.B

k7 k-7 +B

k3 E

E.A.B k2

+A

k-1

k5

k4 E-P

E-P.Q

k-2

k-9

E.A

A B P Q I

+I

k-8

k8

IMP NAD+ XMP NADH inhibitor

E +P k-6

+I

+B

k1

H2O

+Q k-4

k-3

k6 E.P +I

k9

k-10

k10

E.P.I

E-P.I

E.I

E.A.I

Figure S13: Assumed kinetic mechanism for the double-inhibitor experiment. [mechanism] reaction S + N ---> P + NH modifiers I ; Catalysis: E + S E.S : E.S + N E.S.N : E.S.N E-P.NH : E-P.NH E-P + NH : E-P ---> E.P : E.P E + P : E-P + N E-P.N : ; Inhibition: E.S + I E.S.I : E-P + I E-P.I : E.P + I E.P.I :

k1 k2 k3 k4 k5 k6 k7

k-1 k-2 k-3 k-4

k8 k9 k10

k-8 k-9 k-10

k-6 k-7

The DynaFit software derived the following algebraic model: N D

v =

[E]0

N

n1 [A][B]

=

D =

(S55)

d1 + d2 [P] + d3 [Q] + d4 [B] + d5 [A]

(S56)

+d6 [P][I] + d7 [Q][P] + d8 [B][P] + d9 [A][I] + d10 [A][Q] +d11 [A][B] + d12 [Q][P][I] + d13 [B][P][I] + d14 [A][Q][I] +d15 [A][B][I] + d16 [A][B][Q] + d17 [A][B]2 32

(S57)

k1 k2 k3 k4 k−1 (k−2 k−3 + k−2 k4 + k3 k4 )

n1

=

(S58)

d1

= 1

d2

=

k−6 k6

(S60)

d3

=

k−2 k−3 k−4 (k k5 −2 k−3 + k−2 k4 + k3 k4 )

(S61)

d4

=

k2 k3 k4 k−1 (k−2 k−3 + k−2 k4 + k3 k4 )

(S62)

d5

=

k1 k−1

(S63)

d6

=

k−6 k10 k6 k−10

(S64)

d7

=

k−2 k−3 k−4 k−6 k5 k6 (k−2 k−3 + k−2 k4 + k3 k4 )

(S65)

d8

=

k2 k3 k4 k−6 k−1 k6 (k−2 k−3 + k−2 k4 + k3 k4 )

(S66)

d9

=

k1 k8 k−1 k−8

(S67)

d10

=

k1 k−2 k−3 k−4 k−1 k5 (k−2 k−3 + k−2 k4 + k3 k4 )

(S68)

(S59)

d11

=

k1 k2 (k−3 k5 k6 + k4 k5 k6 + k3 k5 k6 + k3 k4 k6 + k3 k4 k5 ) k−1 k5 k6 (k−2 k−3 + k−2 k4 + k3 k4 )

(S69)

d12

=

k−2 k−3 k−4 k−6 k10 k5 k6 k−10 (k−2 k−3 + k−2 k4 + k3 k4 )

(S70)

d13

=

k2 k3 k4 k−6 k10 k−1 k6 k−10 (k−2 k−3 + k−2 k4 + k3 k4 )

(S71)

d14

=

k1 k−2 k−3 k−4 k8 k−1 k5 k−8 (k−2 k−3 + k−2 k4 + k3 k4 )

(S72)

d15

=

k1 k2 k3 k4 (k6 k9 k−10 + k5 k−9 k10 ) k−1 k5 k6 k−9 k−10 (k−2 k−3 + k−2 k4 + k3 k4 ) 33

(S73)

d16

=

k1 k2 k−4 (k−3 + k3 ) k−1 k5 (k−2 k−3 + k−2 k4 + k3 k4 )

(S74)

d17

=

k1 k2 k3 k4 k7 k−1 k5 k−7 (k−2 k−3 + k−2 k4 + k3 k4 )

(S75)

Derived kinetic constants

kcat =

n1 d11

=

k3 k4 k5 k6 k−3 k5 k6 + k4 k5 k6 + k3 k5 k6 + k3 k4 k6 + k3 k4 k5

(S76)

Km(A) =

d4 d11

=

k3 k4 k5 k6 k1 (k−3 k5 k6 + k4 k5 k6 + k3 k5 k6 + k3 k4 k6 + k3 k4 k5 )

(S77)

Km(B) =

d5 d11

=

k5 k6 (k−2 k−3 + k−2 k4 + k3 k4 ) k2 (k−3 k5 k6 + k4 k5 k6 + k3 k5 k6 + k3 k4 k6 + k3 k4 k5 )

(S78)

n1 kcat = Km(A) d4

= k1

kcat n1 = Km(B) d5

=

(S79)

k2 k3 k4 k−2 k−3 + k−2 k4 + k3 k4

34

(S80)

Ki(I,P) =

d2 d6

=

k−10 k10

(S81)

Ki(I,A) =

d5 d9

=

k−8 k8

(S82)

d11 d15

=

k−9 k−10 (k−3 k5 k6 + k4 k5 k6 + k3 k5 k6 + k3 k4 k6 + k3 k4 k5 ) k3 k4 (k6 k9 k−10 + k5 k−9 k10 )

(S83)

Ki(A) =

d1 d5

=

k−1 k1

(S84)

Ki(B) =

d1 d4

=

k−1 (k−2 k−3 + k−2 k4 + k3 k4 ) k2 k3 k4

(S85)

Ki(B,A,Q) =

d10 d16

=

k−2 k−3 k2 (k−3 + k3 )

(S86)

Ki(B,A) =

d11 d17

=

k−7 (k−3 k5 k6 + k4 k5 k6 + k3 k5 k6 + k3 k4 k6 + k3 k4 k5 ) k3 k4 k6 k7

(S87)

d1 d3

=

k5 (k−2 k−3 + k−2 k4 + k3 k4 ) k−2 k−3 k−4

(S88)

d11 d16

=

k−3 k5 k6 + k4 k5 k6 + k3 k5 k6 + k3 k4 k6 + k3 k4 k5 k−4 k6 (k−3 + k3 )

(S89)

d1 d2

=

k6 k−6

(S90)

Ki(I,A,B) =

Ki(Q) = Ki(Q,A,B) = Ki(P) =

Rate equation cast in composite kinetic constants The following algebraic model was derived by algebraic rearrangements of the computergenerated model listed above: N D

v =

[E]

N

kcat(f)

=

35

[A][B] Ki(B) Km(A)

(S91) (S92)

D =

1+

[Q] [B] [A] [P] + + + Ki(P) Ki(Q) Ki(B) Ki(A)

+

[P][I] [Q][P] [B][P] [A][I] + + + Ki(P) Ki(I,P) Ki(Q) Ki(P) Ki(B) Ki(P) Ki(A) Ki(I,A)

+

[A][Q] [A][B] [Q][P][I] + + Ki(A) Ki(Q) Ki(B) Km(A) Ki(Q) Ki(P) Ki(I,P)

+

[B][P][I] [A][Q][I] [A][B][I] + + Ki(B) Ki(P) Ki(I,P) Ki(A) Ki(Q) Ki(I,A) Ki(B) Km(A) Ki(I,A,B)

+

[A][B][Q] [A][B]2 + Ki(B) Km(A) Ki(Q,A,B) Ki(B) Km(A) Ki(B,A)

(S93)

Derivation of the apparent inhibition constant Applying the method of Morrison [17], in the general case when NADH could act as product inhibitor ([Q] > 0), the apparent inhibition constant for I is obtained as shown in Eqn (S94). Ki∗

=

D0 γ

γ

=

[A] [Q][P] [P] + + Ki(P) Ki(I,P) Ki(A) Ki(I,A) Ki(Q) Ki(P) Ki(I,P) +

D0

(S94)

where

[B][P] [A][Q] [A][B] + + Ki(B) Ki(P) Ki(I,P) Ki(A) Ki(Q) Ki(I,A) Ki(B) Km(A) Ki(I,A,B)

= 1+

(S95)

[P] [Q] [B] [A] + + + Ki(P) Ki(Q) Ki(B) Ki(A)

+

[Q][P] [B][P] [A][Q] + + Ki(Q) Ki(P) Ki(B) Ki(P) Ki(A) Ki(Q)

+

[A][B] [A][B][Q] [A][B]2 + + Ki(B) Km(A) Ki(B) Km(A) Ki(Q,A,B) Ki(B) Km(A) Ki(B,A)

(S96)

In the special case of an experiment where NADH is absent in the assay mixture, algebraic terms with [Q] vanish from the definition of the apparent inhibition constant, in which case D0 and γ simplify as follows:

36

D0

=

1+

+ γ

=

[P] [B] [A] [B][P] + + + Ki(P) Ki(B) Ki(A) Ki(B) Ki(P)

[A][B] [A][B]2 + Ki(B) Km(A) Ki(B) Km(A) Ki(B,A)

(S97)

[P] [A] + Ki(P) Ki(I,P) Ki(A) Ki(I,A) +

[A][B] [B][P] + Ki(B) Ki(P) Ki(I,P) Ki(B) Km(A) Ki(I,A,B)

(S98)

Heuristic simulations Based on the kinetic scheme in Figure S13, we have simulated a collection of theoretical curves representing the dependence of the apparent inhibition constant Ki∗ on the presumed relative binding affinities of I toward the covalent intermediate and toward the noncovalent complex, assuming that both types of binding phenomena can occur at the same time. The requisite DynaFit simulation script is shown in Appendix A.2.7. We assumed that Ki(I,A) was 0.7 µM, Ki(I,A,B) was 60 nM, thus assuming that I binds to covalent intermediate with high affinity. The inhibition constant Ki(I,P) was varied from 10 nM to 5.12 µM. The results are shown in Figure S14. Ki(I,A) = 0.7, Ki(I,A,B) = 0.06 µM

0.2

Ki(I,P) = 1.24 Ki(I,P) = 0.64 Ki(I,P) = 0.32

, µM

Ki(I,P) = 0.08 Ki(I,P) = 0.04 uM

0.15

0

0.1

0.1

Ki

(app)

0.2

values

Ki(I,P) = 1.24 Ki(I,P) = 0.64 Ki(I,P) = 0.32

Ki(I,P) = 0.02 Ki(I,P) = 0.01 uM

Ki

(app)

Ki(I,P) = 5.12 Ki(I,P) = 2.56

Ki(I,P) = 0.16 Ki(I,P) = 0.08 Ki(I,P) = 0.04

(app)

, µM

0.3

Linear fit of simulated Ki Ki(I,P) = 5.12 Ki(I,P) = 2.56

0

1000

2000

[P], µM

(a)

0

500

1000

[XMP], µM

(b)

Figure S14: (a) Simulation of Ki∗ values in dependence on the concentrations of [P]. (b) Linear least squares fit of simulated dependence of Ki∗ vs. [P]. The assumed “true” value of Ki(I,A,B) , which measures inhibitor binding to the covalent intermediate E-P, was 60 nM throughout the simulation. The assumed “true” value of Ki(I) , which measures inhibitor binding to the noncovalent complex E·P varied from 40 nM to 5.12 µM and is shown in the margin. The results show that the dependence of Ki∗ on [P] is complex. The plots have various shapes 37

depending on the assumed value of Ki(I,P) . • If inhibitor binding to the covalent intermediate E-P were stronger than the binding to the noncovalent product complex E.P, the apparent inhibition Ki∗ would increase with [P]. The trend changes from approximately hyperbolic to linear. • If the binding to the covalent intermediate E-P were weaker than the binding to the noncovalent product complex E.P, the apparent inhibition Ki∗ would decrease with [P]. The trend would be approximately hyperbolic. • In the range approximately between Ki(I,P) = 80 nM and Ki(I,P) = 160 nM, the plot of Ki∗ vs. [P] would appear approximately horizontal. Linear fit of simulated Ki∗ values The plots of simulated Ki∗ vs. [P] displayed in Figure S14A are clearly nonlinear. However, in an actual experiment the nonlinearity might be somewhat obscured by inevitable experimental error. Once random error is introduced it seems reasonable to assume that experimentally observed plots of Ki∗ vs. [P] might appear essentially linear. It is therefore of interest to perform linear fit of at least some relevant simulated values displayed in Figure S14A. The requisite DynaFit script is listed in Appendix A.2.8. The results of fit are shown in Figure S14B. Recall that the assumed value of the inhibition constant Ki(I,A,B) , which measures the binding affinity of A110 toward the covalent intermediate E-P is 60 nM. Figure S14B shows the various outcomes simulated at values of the inhibition constant Ki(I,P) , which measures the binding affinity of A110 toward the noncovalent complex E.P. The most salient features of the linear regression analyses match the the pattern described above, in the sense that the slope of the regression line depends on the relative binding affinities of I toward the covalent intermediate vs. the noncovalent complex. Most importantly if I binds preferentially to the covalent intermediate, the slope of the regression line is positive. Comparison with experiment Experimentally observed values of the apparent inhibition constant for A110 under relevant experimental conditions are listed in Table S8 above. Those values were subjected to linear regression analysis to determine the experimental slope of the plot of Ki∗ vs. [P]. The requisite DynaFit script is listed in Appendix A.2.9. The script performs a weighted regression using the linear model, Ki∗ = Ki(0) + A × [XMP] where Ki(0) is the value of Ki∗ at zero [XMP] concentration and and A is the slope. The weighting scheme uses the obviously unequal experimental errors associated with each particular measurement of Ki∗ at at different values of [XMP]. The results of fit are shown graphically in Figure 4A (main manuscript). The best fit values are are Ki∗ = (0.137 ± 0.006) µM and A = (7.0 ± 0.6) × 10−5 (a dimensionless value). A comparison between the experimentally observed slope with slopes simulated at various assumed values of the inhibition constant Ki(I,P) is shown in Table S9. The experimentally observed slope is inserted between two immediately adjacent slope values. The results summarized in Table S9 would seem to lead to the tentative conclusion that that the most probable value of Ki(I,P) , which measures the binding affinity of A110 toward the noncovalent complex E.P, lies in the vicinity of 0.5 µM (between 0.32 and 0.64 µM). This value is 38

Ki(I,P) , µM 5.12 2.56 1.28 0.64 experiment 0.32 0.16

slope A × 105 ± Std. Error 11.3 10.8 9.8 8.2 7.0 5.5 1.9

± ± ± ± ± ± ±

0.1 0.1 0.1 0.1 0.6 0.1 0.1

Table S9: Comparison of experimentally observed slope of the plot of Ki∗ vs. [P] (Figure S14B) with the slopes simulated at various assumed values of the inhibition constant Ki(I,P) . approximately ten fold higher than the inhibition constant Ki(I,A,B) = 60 nM, which measures the the binding affinity of A110 toward the covalent intermediate. 3.4. Steady-state kinetics: Summary and conclusions Based on the results of the combined initial-rate experiments, we can arrive at the following mechanistic conclusions: 1. Under IMP saturating conditions, the uncompetitive inhibition constant of A110, defined in terms of microscopic rate constants by Eqn (S29), is Ki(I,B) ≈ 50 nM. This suggests strong binding to the covalent intermediate E-P or to the noncovalent enzyme–product complex E.P. 2. The double-inhibitor experiment conducted under IMP sub-saturation suggests that the inhibitor binds preferentially to the covalent intermediate. A semi-quantitative assessment suggest approximately ten-fold stronger binding, as measured by the inhibition constant. 3. The uncompetitive inhibition constant Ki(I,B) is not by definition equal to the dissociation complex of the requisite ternary complex, contrary to generalized suggestions in textbook literature regarding the nature of steady-state inhibition constants. 4. The inhibitor also binds relatively weakly (Ki(I) = Kd(EA,I) ≈ 0.7 µM) to the enzyme– substrate (IMP) complex. In this case the inhibition and dissociation constants are by definition identical. 5. Thus, under IMP saturating conditions, in the conventional nomenclature [25] A110 would be characterized as a “mixed-type, predominantly uncompetitive” inhibitor. 6. The substrate kinetic properties of BaIMPDH∆L are well reproduced in the stopped-flow vs. initial rate data. This includes quantitative measures of (a) the saturation behavior; (b) substrate inhibition by NAD+ ; and (c) product inhibition by NADH. 4. Responses to Reviewers 4.1. Linear vs. logarithmic plots A Reviewer has raised a question regarding the relative merits of linearly scaled vs. logarithmically scaled time axis, in plotting stopped-flow transient kinetic data. One particular suggestion by the Reviewer was that perhaps plotting stopped-flow data in linear as opposed to logarithmic time increases our ability to detect any possible “lag” phase transients. 39

This gives us an opportunity to demonstrate that “lag” type transients are extremely difficult to detect by naked eye simply by examining the plots of experimental data and the overlayed model curves, regardless of which type of scaling (linear or logarithmic) is used. However, an exquisitely informative alternative is presented by plotting instantaneous reaction rates vs. time, again regardless of which type of scaling is used on the time axis. [NADH] = 0, [A110] = 0

400

d(1000 ∆A340) / dt

200

0 0.5 0 -0.5

0

residuals

0.25 0.5 1 2 4 6 8 mM

600

40

[NADH] = 0, [A110] = 0

20

1000 ∆A340

60

80

0.25 0.5 1 2 4 6 8 mM

0

0.2

0.4

0

0.2

0.4

time, s

time, s

(a)

(b)

(c)

(d)

Figure S15: (a) The upper-left panel in Figure S2 above, redrawn in linear time scale. (b) Corresponding instantaneous rate plots. (c) Magnified portion of the lowest concentration curve, [NAD] = 0.25 mM, showing the slight “lag” transient. (d) Corresponding instantaneous rate plot. Figure S15a displays what was shown previously as the upper-left panel in Figure S2 above, this time with linearly scaled time. Figure S15b shows the corresponding instantaneous rate plots. Recall that, at an arbitrary reaction time t, the “instantaneous rate” is the slope of a tangent 40

line drawn with respect to the corresponding reaction progress curve. Most importantly, any observed increase in the instantaneous rate curve represent a “lag” phase being present in the underlying theoretical model curve. Please note that only an extremely well-trained eye could possibly detect a very slight “lag”, occurring with the first approximately 50 milliseconds especially in the low-concentration progress curves shown Figure S15a. Now note that this lag is rendered prominently visible in the accompanying instantaneous rate plots, Figure S15b. All instantaneous rate curves in Figure S15b display an increase in the reaction rate between time t = 0 and t = 30 msec. A visual verification of the very subtle but statistically significant “lag” is presented in Figure S15c and Figure S15d. The “lag” phase becomes visually detectable in the data plot (as opposed to the instantaneous rate plot) only by “zooming” very purposely on the initial approximately 10% section (the first 50 msec of the total 500 msec trace). In contrast, the instantaneous rate plot shows a very clear “lag” (i.e., increase in the observed instantaneous rate) for all progress curves plotted in their entirety. [NADH] = 0, [A110] = 6 µM

[NADH] = 0, [A110] = 6 µM

d(1000 ∆A340) / dt

400

600

0.25 0.5 1 2 4 6 8 mM

200

20 0

1000 ∆A340

40

0.25 0.5 1 2 4 6 8 mM

0 -1

0

residuals

0.5 -0.5 -1.5

0

1

2

0

1

2

time, s

time, s

(a)

(b)

Figure S16: (a) The lower-left panel in Figure S2 above, redrawn in linear time scale. (b) Corresponding instantaneous rate plots. This Reviewer also suggested that the “shoulder” feature between t = 0.1 sec and t = 0.5 sec, displayed in the lower left panel of Figure S2 and discussed in section 2.3.1, could be easily visualized as a simple approach to “steady-state”, if plotted in the linear as opposed to logarithmic time coordinates. Figure S16 helps explain that this statement is only partially true. As a matter of fact, several curves shown in Figure S16 (in particular [NAD] = 0.25 and 0.5 mM) never actually reach “steady-state”. Note that the [NAD] = 0.25 and 0.5 mM curves never develop a truly linear phase corresponding to genuine steady-state. The reason is that in the presence of the inhibitor at [I] = 6 µM there exists a complex interplay between a large number competing processes, as befits a reaction mechanism characterized by thirteen microscopic rate constants. 41

References [1] P. Kuzmiˇc, Program DYNAFIT for the analysis of enzyme kinetic data: Application to HIV proteinase, Anal. Biochem. 237 (1996) 260–273. [2] P. Kuzmiˇc, DynaFit - A software package for enzymology, Meth. Enzymol. 467 (2009) 247–280. [3] J. M. Beechem, Global analysis of biochemical and biophysical data, Meth. Enzymol. 210 (1992) 37–54. [4] A. Raue, C. Kreutz, T. Maiwald, J. Bachmann, M. Schilling, U. Klingmüller, J. Timmer, Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood, Bioinformatics 25 (2009) 1923–1929. [5] T. V. Riera, W. Wang, H. R. Josephine, L. Hedstrom, A kinetic alignment of orthologous inosine-5’-monophosphate dehydrogenases, Biochemistry 47 (2008) 8689–8696. [6] L. Hedstrom, IMP dehydrogenase: structure, mechanism, and inhibition, Chem. Rev. 109 (2009) 2903–2928. [7] K. V. Price, R. M. Storm, J. A. Lampinen, Differential Evolution - A Practical Approach to Global Optimization, Springer Verlag, Berlin - Heidelberg, 2005. [8] M. L. Johnson, Use of least-squares techniques in biochemistry, Meth. Enzymol. 240 (1994) 1–22. [9] D. M. Bates, D. G. Watts, Nonlinear Regression Analysis and its Applications, Wiley, New York, 1988. [10] I. Brooks, D. Watts, K. Soneson, P. Hensley, Determining confidence intervals for parameters derived from analysis of equilibrium analytical ultracentrifugation data, Meth. Enzymol. 240 (1994) 459–78. [11] D. G. Watts, Parameter estimation from nonlinear models, Methods Enzymol. 240 (1994) 24–36. [12] K. A. Johnson, Z. B. Simpson, T. Blom, Global Kinetic Explorer: A new computer program for dynamic simulation and fitting of kinetic data, Anal. Biochem. 387 (2009) 20–29. [13] K. A. Johnson, Z. B. Simpson, T. Blom, FitSpace Explorer: An algorithm to evaluate multidimensional parameter space in fitting kinetic data, Anal. Biochem. 387 (2009) 30–41. [14] K. A. Johnson, Fitting enzyme kinetic data with KinTek Global Kinetic Explorer, Meth. Enzymol. 267 (2009) 601–626. [15] H. Bisswanger, Enzyme Kinetics, 2nd Edition, Wiley-VCH, Tuebingen, 2008. [16] I. H. Segel, Enzyme Kinetics, Wiley, New York, 1975. [17] J. F. Morrison, Kinetics of the reversible inhibition of enzyme-catalysed reactions by tightbinding inhibitors, Biochim. Biophys. Acta 185 (1969) 269–286. 42

[18] E. L. King, C. Altman, A schematic method of deriving the rate laws for enzyme-catalyzed reactions, J. Phys. Chem. 60 (1956) 1375–1378. [19] W. Cleland, The kinetics of enzyme-catalyzed reactions with two or more substrates or products: I. nomenclature and rate equations, Biochim. Biophys. Acta 67 (1963) 104–137. [20] S. Cha, Tight-binding inhibitors. i. kinetic behavior, Biochem. Pharmacol. 24 (1975) 2177– 2185. [21] J. W. Williams, J. F. Morrison, The kinetics of reversible tight-binding inhibition, Meth. Enzymol. 63 (1979) 437–467. [22] D. H. Rich, D. B. Northrop, Enzyme kinetics in drug design: Implications of multiple forms of enzyme on substrate and inhibitor structure-activity correlations, in: T. J. Perun, C. L. Prost (Eds.), Computer-Aided Drug Design: Methods and Applications, Marcel Dekker, New York, 1989, pp. 185–250. [23] D. Northrop, Fitting enzyme-kinetic data to v/k, Anal. Biochem. 321 (1983) 457–461. [24] P. R. Bevington, Data Reduction and Error Analysis in the Physical Sciences, McGraw-Hill, New York, 1969. [25] International Union of Biochemistry (IUB), Symbolism and terminology in enzyme kinetics, Biochem. J. 213 (1981) 561–571.

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Appendix A. DynaFit scripts A.1. Global fit of stopped-flow data This DynaFit [2] script performs global fit of combined transient kinetic data using one of the 16 combinatorial replicates (replicate no. 1 listed in Table S2). The double question marks (??) identify the model parameters to be subjected to the full asymmetric confidence interval search. [task] task = fit data = progress [mechanism] E.A + B E.A.B : k2 k-2 E.A.B E.P.Q : k3 k-3 E.P.Q E.P + Q : k4 k-4 E.P ---> E.A + P : k5 E.P + B E.P.B : k7 k-7 E.A + I E.A.I : k8 k-8 E.P + I E.P.I : k9 k-9 [constants] k2 = 0.01 ?? , k-2 = 10 ?? k3 = 100 ?? , k-3 = 10 ?? k4 = 1000 ?? , k-4 = 10 ?? k5 = 10 ?? k7 = 0.01 ?? , k-7 = 100 ?? k8 = 10 ?? , k-8 = 10 ?? k9 = 1 ?? , k-9 = 0.1 ?? [parameters] In = 6 ? [concentrations] E.A = 2.5 ? [responses] Q = 6.22 E.P.Q = 1 * Q [data] directory ./project/impdh/01/progress/data monitor E.A, E.A.B, E.P.Q, E.P, E.P.B, E.A.I, E.P.I plot logarithmic ; ; Control experiment -- DMSO only: ; graph [NADH] = 0, [A110] = 0 maximum 0.5 sheet N1.csv shift 0 | column 2 | offset auto ? | conc B = 250 | shift 1 | column 3 | offset auto ? | conc B = 500 | shift 2 | column 4 | offset auto ? | conc B = 1000 | shift 4 | column 5 | offset auto ? | conc B = 2000 |

44

label label label label

0.25 0.5 1 2

shift 12 | column 6 | offset auto ? | conc B = 4000 | label 4 shift 20 | column 7 | offset auto ? | conc B = 6000 | label 6 shift 28 | column 8 | offset auto ? | conc B = 8000 | label 8 mM ; ; Product inhibition by NADH (60 uM): ; graph [NADH] = 60 {/Symbol m}M, [A110] = 0 maximum 0.5 sheet H1.csv shift 0 | column 2 | offset -373 ? | conc Q = 60, B = 250 | label 0.25 shift 1 | column 3 | offset -372 ? | conc Q = 60, B = 500 | label 0.5 shift 2 | column 4 | offset -371 ? | conc Q = 60, B = 1000 | label 1 shift 4 | column 5 | offset -369 ? | conc Q = 60, B = 2000 | label 2 shift 12 | column 6 | offset -361 ? | conc Q = 60, B = 4000 | label 4 shift 20 | column 7 | offset -353 ? | conc Q = 60, B = 6000 | label 6 shift 28 | column 8 | offset -345 ? | conc Q = 60, B = 8000 | label 8 mM ; ; Inhibition by A110 (6 uM): ; graph [NADH] = 0, [A110] = 6 {/Symbol m}M maximum off sheet I1.csv shift 0 | column 2 | offset auto ? | conc I = 1 * In, B = 250 | label 0.25 shift 1 | column 3 | offset auto ? | conc I = 1 * In, B = 500 | label 0.5 shift 2 | column 4 | offset auto ? | conc I = 1 * In, B = 1000 | label 1 shift 4 | column 5 | offset auto ? | conc I = 1 * In, B = 2000 | label 2 shift 6 | column 6 | offset auto ? | conc I = 1 * In, B = 4000 | label 4 shift 10 | column 7 | offset auto ? | conc I = 1 * In, B = 6000 | label 6 shift 12 | column 8 | offset auto ? | conc I = 1 * In, B = 8000 | label 8 mM [output] directory ./project/impdh/01/progress/output/fit-N1H1I1 [settings] {Filter} ZeroBaselineSignal = y XMin = 0.004 TimeFirstMesh = 0.003 PointsPerDataset = 25 ExponentialSpacing = y {Output} WriteTeX = y WriteEPS = y XAxisLabel = time, s YAxisLabel = {/Symbol D}A_{340}, mOD {ConfidenceIntervals} SquaresIncreasePercent = 5 [end]

45

A.2. Global fit of initial rates A.2.1. Predict kinetic constants: “Classical” rate equation This DynaFit [2] script performs global fit of combined initial rate data to the “classical” rate equation automatically derived by the King-Altman method. All microscopic rate constants are held fixed at values determined by the analysis of stopped-flow data. Only the enzyme active site concentration is optimized. Any “tight binding” is ignored. Compare observed initial rates with those predicted from the minimal transient kinetic model based on stopped-flow data. King-Altman approximation: any "tight binding" is ignored. ;______________________________________________________________________ [task] task = fit data = rates approximation = king-altman [mechanism] enzyme E.A reaction B ---> P + Q modifiers I E.A + B E.A.B : k2 k-2 E.A.B E.P.Q : k3 k-3 E.P.Q E.P + Q : k4 k-4 E.P ---> E.A + P : k5 E.P + B E.P.B : k7 k-7 E.A + I E.A.I : k8 k-8 E.P + I E.P.I : k9 k-9 [constants] k2 = 0.0318 , k-2 = 10.9 k3 = 82.1 , k-3 = 44.3 k4 = 11500 , k-4 = 100 k5 = 13.6 k7 = 100 , k-7 = 566000 k8 = 13.9 , k-8 = 11 k9 = 5.51 , k-9 = 0.27 [responses] Q = 6.22 [parameters] cE1 = 0.01 ? cE2 = 0.01 ? cE3 = 0.01 ? cE4 = 0.01 ? cE5 = 0.01 ? [data] variable B directory ./project/IMPDH/01/rates/data graph NAD sheet nad-b.csv column 2 | conc E.A = 1 * cE1 | label A = 750 uM column 3 | conc E.A = 1 * cE1 | label A = 1000 (a)

46

sheet nad.csv column 3 | conc E.A = 1 * cE1 | label A = 1000 (b) column 2 | conc E.A = 1 * cE2 | label A = 1000 (c) graph NAD + NADH sheet nad-nadh.csv column 2 | conc E.A = 1 * cE3, Q = 0 | label Q = column 3 | conc E.A = 1 * cE3, Q = 50 | label Q = column 4 | conc E.A = 1 * cE3, Q = 100 | label Q = column 5 | conc E.A = 1 * cE3, Q = 150 | label Q = graph NAD (low) + A110 sheet nad-a110-d.csv column 2 | conc E.A = 1 * cE4, I = 0.000 | label I column 3 | conc E.A = 1 * cE4, I = 0.015 | label I column 4 | conc E.A = 1 * cE4, I = 0.030 | label I column 5 | conc E.A = 1 * cE4, I = 0.060 | label I column 6 | conc E.A = 1 * cE4, I = 0.120 | label I column 7 | conc E.A = 1 * cE4, I = 0.180 | label I graph NAD (high) + A110 sheet nad-a110-a.csv column 2 | conc E.A = 1 * cE5, I = 0.000 | label I column 3 | conc E.A = 1 * cE5, I = 0.015 | label I column 4 | conc E.A = 1 * cE5, I = 0.030 | label I column 5 | conc E.A = 1 * cE5, I = 0.060 | label I column 6 | conc E.A = 1 * cE5, I = 0.090 | label I column 7 | conc E.A = 1 * cE5, I = 0.180 | label I [output] directory ./project/IMPDH/01/rates/output/fit-micro [settings] {Filter} XMax = 7000 {Output} WriteEPS = y WriteTeX = y ResidualsEPS = n XAxisLabel = [NAD^+], {/Symbol m}M YAxisLabel = v, mOD/min [end]

0 uM 50 100 150

= = = = = =

0 uM 0.015 0.030 0.060 0.120 0.180

= = = = = =

0 uM 0.015 0.030 0.060 0.090 0.150

A.2.2. Predict kinetic constants: “Tight binding” rate equation This DynaFit [2] script performs global fit of combined initial rate data to the “Morrison equation” defined in Eqns (S40)–(S42). All kinetic constants are held fixed at values determined by the analysis of stopped-flow data. Only the enzyme active site concentration is optimized. The model is formulated in terms of composite kinetic constants. “Tight binding” is properly accounted for. Compare observed initial rates with those predicted from the minimal transient kinetic model based on stopped-flow data. ;______________________________________________________________________ [task] task = fit

47

data = generic [parameters] E, B, Q, I kcat, KmB, KiI, KiIB, KiB, KiQ, KiQB rQ cE1, cE2, cE3, cE4, cE5 [model] ; ; Enzyme concentrations in different experiments: All optimized. ; cE1 = 0.01 ? cE2 = 0.01 ? cE3 = 0.01 ? cE4 = 0.01 ? cE5 = 0.01 ? ; ; Extinction coefficient of NADH: ; rQ = 6.22 ; fixed ! ; ; Predicted kinetic constants: All fixed in the regression! ; kcat = 11.6 KmB = 415 KiI = 0.791 KiIB = 0.0572 KiB = 6610 KiQ = 301 KiQB = 87.2 ; ; "Morrison equation" (uM/sec) for mechanism at saturating [IMP]: ; n = B/KmB d = 1 + Q/KiQ + B/KmB*(1 + Q/KiQB + B/KiB) alpha = 1/KiI*(1 + Q/KiQ) + B/(KmB*KiIB) Kiapp = d/alpha e = E - I - Kiapp v = kcat * n/d * (e + sqrt (e*e + 4*Kiapp*E))/2 ; ; Observed rate (mOD/sec): ; vObs = v * rQ [data] variable B directory ./project/IMPDH/01/rates/data graph NAD sheet nad-b.csv column 2 | param E = 1 * cE1 | label A = 0.75 mM column 3 | param E = 1 * cE1 | label A = 1.0 (a) sheet nad.csv column 3 | param E = 1 * cE1 | label A = 1.0 (b)

48

column 2 | param E = 1 * cE2 | label A = 1.0 (c) graph NAD + NADH sheet nad-nadh.csv column 2 | param E = 1 * cE3, Q = 0 | label Q = 0 mM column 3 | param E = 1 * cE3, Q = 50 | label Q = 0.05 column 4 | param E = 1 * cE3, Q = 100 | label Q = 0.10 column 5 | param E = 1 * cE3, Q = 150 | label Q = 0.15 graph NAD (low) + A110 sheet nad-a110-d.csv column 2 | param E = 1 * cE4, I = 0.000 | label I = 0 nM column 3 | param E = 1 * cE4, I = 0.015 | label I = 15 column 4 | param E = 1 * cE4, I = 0.030 | label I = 30 column 5 | param E = 1 * cE4, I = 0.060 | label I = 60 column 6 | param E = 1 * cE4, I = 0.120 | label I = 120 column 7 | param E = 1 * cE4, I = 0.180 | label I = 180 graph NAD (high) + A110 sheet nad-a110-a.csv column 2 | param E = 1 * cE5, I = 0.000 | label I = 0 nM column 3 | param E = 1 * cE5, I = 0.015 | label I = 15 column 4 | param E = 1 * cE5, I = 0.030 | label I = 30 column 5 | param E = 1 * cE5, I = 0.060 | label I = 60 column 6 | param E = 1 * cE5, I = 0.090 | label I = 90 column 7 | param E = 1 * cE5, I = 0.180 | label I = 150 [output] directory ./project/IMPDH/01/rates/output/compare-v [settings] {Filter} XMax = 7000 {ConfidenceIntervals} LevelPercent = 90 {Output} WriteTeX = y ResidualsEPS = n XAxisLabel = [NAD^+], {/Symbol m}M YAxisLabel = v, mOD/sec [end]

A.2.3. Kinetic isotope effect This DynaFit [2] script performs global fit of combined 1 H-IMP and 2 D-IMP data to the steady-state rate equation that takes into account substrate inhibition by NAD+ . The enzyme concentration is assumed to be identical in both experiments. Fit H/D isotope effect data. Show 1H-IMP and 2D-IMP data in the same graph, but assign to both curves unique kcat and Km values. Use Nothrop’s 1983 equation where "kK = kcat/Km" is a parameter. ;______________________________________________________________________ [task] task = fit data = generic [parameters]

49

S Km, Ki, kK, Eo, rP [model] rP = 6.22 Eo = 0.010 v = rP * Eo * kK * Km * S/(Km + S*(1 + S/Ki)) [data] variable S directory ./project/impdh/01/rates/data sheet nad-isotope.csv column 2 | label ^{1}H-IMP param Km = 500 ??, Ki = 5000 ??, kK = 0.01 ?? column 3 | label ^{2}H-IMP param Km = 500 ??, Ki = 5000 ??, kK = 0.01 ?? [output] directory ./project/impdh/01/rates/output/fit-isotope [settings] {Output} WriteEPS = yes ResidualsEPS = no WriteTeX = yes XAxisLabel = [NAD^+], {/Symbol m}M YAxisLabel = v, mOD/sec [end]

A.2.4. Apparent Michaelis constant of IMP This DynaFit [2] script used to determine the apparent Michaelis constant of IMP at [NAD+ ] = 1.5 mM. Fit IMP saturation data at NAD+ = 1200 to determine the apparent Km for IMP at this NAD+ concentration. ;______________________________________________________________________ [task] task = fit data = generic [parameters] S, Vmax, Km [model] Vmax = 0.5 ? Km = 50 ? v = Vmax * S/(S + Km) [data] variable S graph IMP : Km^{(app)} = (61 +/- 3) {/Symbol m}M set rates [output] directory ./project/impdh/01/rates/output/fit-475 [settings] {Output} XAxisLabel = [IMP], {/Symbol m}M

50

YAxisLabel = v, mOD/sec ResidualsEPS = n WriteTeX = y [set:rates] IMP rate 10 0.0736 20 0.1308 30 0.1702 50 0.2327 75 0.2593 100 0.3098 150 0.3617 200 0.3837 300 0.4192 500 0.4493 750 0.4545 1000 0.4863 [end]

A.2.5. Apparent inhibition constant of XMP This DynaFit [2] script used to determine the apparent inhibition constant of XMP at [NAD+ ] = 1.5 mM and [IMP] = 200 µM. Determine apparent Ki for XMP at NAD+ = 1500 uM and IMP = 200 uM. ;______________________________________________________________________ [task] task = fit data = generic [parameters] P, Vo, Ki [model] Vo = 0.8 ? Ki = 1000 ? v = Vo * Ki / (P + Ki) [data] variable P graph XMP : Ki^{(app)} = (1160 +/- 50) {/Symbol m}M set rates [output] directory ./project/impdh/01/rates/output/fit-476 [settings] {Output} XAxisLabel = [XMP], {/Symbol m}M YAxisLabel = v, mOD/sec ResidualsEPS = n WriteTeX = y IncludeYZero = y [set:rates] XMP rate 0 0.8958

51

125 0.8009 250 0.7491 500 0.6407 1000 0.4744 2000 0.3232 [end]

A.2.6. Apparent inhibition constant A110 in dependence on [XMP] This DynaFit [2] script used to determine the apparent inhibition constant of A110 at various fixed concentrations of [XMP] ([NAD+ ] = 1.5 mM and [IMP] = 200 µM). Apparent inhibition constants for A110 vs. XMP [task] task = fit data = generic [parameters] I, E, K, Vo [model] E = 0.01 e = E - I - K v = Vo*(e + sqrt(e*e + 4*E*K))/(2*E) [data] directory ./project/impdh/01/rates/data variable I graph A110 : K_i^{(app)} vs. [XMP] sheet a110-xmp.csv column 2 | param Vo = 1 ?, K = 0.05 ? | label P = column 3 | param Vo = 1 ?, K = 0.05 ? | label P = column 4 | param Vo = 1 ?, K = 0.05 ? | label P = column 5 | param Vo = 1 ?, K = 0.05 ? | label P = column 6 | param Vo = 1 ?, K = 0.05 ? | label P = column 7 | param Vo = 1 ?, K = 0.05 ? | label P = [output] directory ./project/impdh/01/rates/output/fit-470 [settings] {Output} IncludeYZero = y WriteTeX = y XAxisLabel = [I], {/Symbol m}M YAxisLabel = v, mOD/sec ResidualsEPS = n [end]

0 uM 125 250 500 1000 2000

A.2.7. Simulate dependence of Ki ∗ for A110 on [XMP] This DynaFit [2] script used to simulate the expected dependence of Ki ∗ for A110 on [XMP]. [task] task = simulate data = generic

52

[parameters] A, B, P ; concentrations KmA, KiB, KiP, KiBA ; substrate kinetic constants KiIA, KiIP, KiIAB ; inhibition constants for "I" [model] A = 200 ; concentrations (P: see [data]) B = 1500 KmA = 50 ; substrate kinetic constants KiA = 140 KiB = 1300 KiP = 220 KiBA = 6500 KiIA = 0.7 ; inhibition constants (KiP: see [data]) KiIAB = 0.06 Do = 1 + P/KiP + B/KiB + A/KiA + B*P/(KiB*KiP) + A*B/(KiB*KmA)*(1 + B/KiBA) g1 = P/(KiP*KiIP) + A/(KiA*KiIA) g2 = B*P/(KiB*KiP*KiIP) + A*B/(KiB*KmA*KiIAB) gamma = g1 + g2 KiApp = Do / gamma [data] variable P mesh logarithmic from 2000 to 10 step 0.5 directory ./project/impdh/kiapp-xmp/data graph K_{i(I,A)} = 0.7, K_{i(I,A,B)} = 0.06 {/Symbol m}M sheet sim-002.csv column 2 | param KiIP = 5.12 | label Ki(I,P) = 5.12 column 3 | param KiIP = 2.56 | label Ki(I,P) = 2.56 column 4 | param KiIP = 1.24 | label Ki(I,P) = 1.24 column 5 | param KiIP = 0.64 | label Ki(I,P) = 0.64 column 6 | param KiIP = 0.32 | label Ki(I,P) = 0.32 column 7 | param KiIP = 0.16 | label Ki(I,P) = 0.16 column 8 | param KiIP = 0.08 | label Ki(I,P) = 0.08 column 9 | param KiIP = 0.04 | label Ki(I,P) = 0.04 column 10 | param KiIP = 0.02 | label Ki(I,P) = 0.02 column 11 | param KiIP = 0.01 | label Ki(I,P) = 0.01 uM [output] directory ./project/impdh/kiapp-xmp/output/sim-002 [settings] {Output} XAxisLabel = [P], {/Symbol m}M YAxisLabel = K_i^{(app)}, {/Symbol m}M ResidualsEPS = n [end]

A.2.8. Linear fit of simulated Ki ∗ vs. [XMP] values This DynaFit [2] script used to perform linear fit of the simulated dependence of Ki ∗ for A110 on [XMP]. Note that that CSV data file being pointed to is the same file that is mentioned in Appendix A.2.7. The parameter a is the slope of the linear fit. [task]

53

task = fit data = generic [parameters] XMP, Ki0, a [model] KiApp = Ki0 + XMP * a [data] directory ./project/impdh/kiapp-xmp/data variable XMP sheet sim-002b.csv graph Linear fit of simulated K_{i}^{(app)} values column 2 | param Ki0 = 0.1 ?, a = 0.0001 ? | label Ki(I,P) column 3 | param Ki0 = 0.1 ?, a = 0.0001 ? | label Ki(I,P) column 4 | param Ki0 = 0.1 ?, a = 0.0001 ? | label Ki(I,P) column 5 | param Ki0 = 0.1 ?, a = 0.0001 ? | label Ki(I,P) column 6 | param Ki0 = 0.1 ?, a = 0.0001 ? | label Ki(I,P) column 7 | param Ki0 = 0.1 ?, a = 0.0001 ? | label Ki(I,P) column 8 | param Ki0 = 0.1 ?, a = -0.0001 ? | label Ki(I,P) [output] directory ./project/impdh/01/rates/output/fit-kiapp-003 [settings] {Filter} XMax = 1000 {Output} XAxisLabel = [XMP], {/Symbol m}M YAxisLabel = K_i^{(app)}, {/Symbol m}M ResidualsEPS = n [end]

= = = = = = =

5.12 2.56 1.24 0.64 0.32 0.08 0.04 uM

A.2.9. Linear fit of experimental Ki ∗ vs. [XMP] values This DynaFit [2] script used to perform weighted linear fit of the experimentally observed dependence of Ki ∗ for A110 on [XMP]. The reciprocal value of the formal standard error of individual Ki ∗ determinations is used as a weighting factor in the regression analysis. [task] task = fit data = generic [parameters] XMP, Ki0, a [model] Ki0 = 0.1 ? a = 0.0001 ? KiApp = Ki0 + XMP * a [data] directory ./project/impdh/01/rates/data variable XMP error data sheet xmp-kiapp.csv graph A110 : K_i^{(app)} vs. [XMP] column 2 error 3 ; error-weighted fit!

54

[output] directory ./project/impdh/01/rates/output/fit-kiapp-001 [settings] {Output} XAxisLabel = [XMP], {/Symbol m}M YAxisLabel = K_i^{(app)}, {/Symbol m}M IncludeYZero = y ResidualsEPS = n [end]

B. Raw data files and DynaFit script files This section describes the raw experimental data files distributed with this report, as well as the distributed input “script” files for the software DynaFit [1, 2] that were used to perform all kinetic analyses. The main purpose is to allow any interested person to reproduce our kinetic analyses and in so doing independently verify the results. A secondary goal is to provide sufficient level of detail that might allow interested parties to reproduce the kinetic results reported here while using other third-party software packages. Both types of data files (raw data and DynaFit scripts) are collected in the ZIP archive file BaIMPDHdL-A110-transient-DynaFit.ZIP, which represents the second of two Supporting Information components, the first component being this PDF file. B.1. Installation instructions In order to reproduce or modify all data kinetic data analyses presented in this report, please follow these steps: 1. Download the most recent version of the DynaFit software package [2] for any particular platform (Mac OS or Windows) from the BioKin website.8 2. Install the software package simply by extracting the downloaded ZIP archive file (e.g., dynafit4-win.zip). This will create a directory named DynaFit4. 3. Automatically generate and install the free academic license, by filling out the online license request form. 9 4. Download and extract the Supporting Information file BaIMPDHdL-A110-transient-DynaFit.ZIP from the ACS journal page. This will create a directory named ./project/IMPDH/01, containing a number of sub-directories as described below. 5. Move or copy the newly extracted directory ./project/IMPDH/01 into the DynaFit main installation directory. The final layout of files and directories must match the schematic diagram below. \---DynaFit4 +---examples | +---help | 8 http://www.biokin.com/dynafit/ 9 http://www.biokin.com/dynafit/license/academic.html

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+---manual | +---project